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Large-scale shell-model analysis of the neutrinoless $ββ$ decay of $^{48}$Ca

We present the nuclear matrix element for the neutrinoless double-beta decay of $^{48}$Ca based on large-scale shell-model calculations including two harmonic oscillator shells ($sd$ and $pf$ shells). The excitation spectra of $^{48}$Ca and $^{48}$Ti, and the two-neutrino double-beta decay of $^{48}$Ca are reproduced in good agreement to experiment. We find that the neutrinoless double-beta decay nuclear matrix element is enhanced by about 30\% compared to $pf$-shell calculations. This reduces the decay lifetime by almost a factor of two. The matrix-element increase is mostly due to pairing correlations associated with cross-shell $sd$-$pf$ excitations. We also investigate possible implications for heavier neutrinoless double-beta decay candidates.