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Energy density functional study of nuclear matrix elements for neutrinoless $ββ$ decay

We present an extensive study of nuclear matrix elements (NME) for the neutrinoless double beta decay of the nuclei $^{48}$Ca, $^{76}$Ge, $^{82}$Se, $^{96}$Zr, $^{100}$Mo, $^{116}$Cd, $^{124}$Sn, $^{128}$Te, $^{130}$Te, $^{136}$Xe, and $^{150}$Nd based on state-of-the-art energy density functional methods using the Gogny D1S functional. Beyond mean-field effects are included within the generating coordinate method with particle number and angular momentum projection for both initial and final ground states. We obtain a rather constant value for the NME's around 4.7 with the exception of $^{48}$Ca and $^{150}$Nd, where smaller values are found. We analyze the role of deformation and pairing in the evaluation of the NME and present detailed results for the decay of $^{150}$Nd.