Neutrinoless $\beta \beta$ decay nuclear matrix elements in an isotopic chain

We analyze nuclear matrix elements (NME) of neutrinoless double beta decay calculated for the Cadmium isotopes. Energy density functional methods including beyond mean field effects such as symmetry restoration and shape mixing are used. Strong shell effects are found associated to the underlying nuclear structure of the initial and final nuclei. Furthermore, we show that NME for two-neutrino double beta decay evaluated in the closure approximation, $M_{\mathrm{c}\mathrm{l}}^{2\nu }$, display a constant proportionality with respect to the Gamow-Teller part of the neutrinoless NME, $M_{\mathrm{G}\mathrm{T}}^{0\nu }$. This opens the possibility of determining the $M_{\mathrm{G}\mathrm{T}}^{0\nu }$ matrix elements from ${\beta }^{\mp }$ Gamow-Teller strength functions. Finally, the interconnected role of deformation, pairing, configuration mixing and shell effects in the NMEs is discussed.