Double Gamow-Teller transitions and its relation to neutrinoless $\beta\beta$ decay

We study the double Gamow-Teller (DGT) strength distribution of $^{48}$Ca with state-of-the-art large-scale nuclear shell model calculations. Our analysis shows that the centroid energy of the DGT giant resonance depends mostly on the isovector pairing interaction, while the resonance width is more sensitive to isoscalar pairing. Pairing correlations are also key in neutrinoless $\beta\beta$ ($0\nu\beta\beta$) decay. We find a simple relation between the centroid energy and width of the $^{48}$Ca DGT giant resonance and %the value of the $0\nu\beta\beta$ decay nuclear matrix element. More generally, we observe a very good linear correlation between the DGT transition to the ground state of the final nucleus and the $0\nu\beta\beta$ decay matrix element. The correlation, which originates on the dominant short-range character of both transitions, extends to heavier systems including several $\beta\beta$ emitters, and also holds in energy-density functional results. Our findings suggest that DGT experiments can be a very valuable tool to obtain information on the value of $0\nu\beta\beta$ decay nuclear matrix elements.