Energy Density Functional Study of Nuclear Matrix Elements for Neutrinoless $\beta \beta$ Decay

We present an extensive study of nuclear matrix elements (NME) for the neutrinoless double-beta decay of the nuclei ${}^{48}\mathrm{Ca}$, ${}^{76}\mathrm{Ge}$, ${}^{82}\mathrm{Se}$, ${}^{96}\mathrm{Zr}$, ${}^{100}\mathrm{Mo}$, ${}^{116}\mathrm{Cd}$, ${}^{124}\mathrm{Sn}$, ${}^{128}\mathrm{Te}$, ${}^{130}\mathrm{Te}$, ${}^{136}\mathrm{Xe}$, and ${}^{150}\mathrm{Nd}$ 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 NMEs around 4.7 with the exception of ${}^{48}\mathrm{Ca}$ and ${}^{150}\mathrm{Nd}$, 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 ${}^{150}\mathrm{Nd}$.

Figure 1

Comparison between theoretical and experimental ground state bands for ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$. Inset: Collective wave functions of the ground states as a function of the intrinsic deformation. The values for ${}^{150}\mathrm{Sm}$ have been shifted up by 0.05.

Figure 2

(a) Strength of the GT operator as a function of the intrinsic deformation of the mother and granddaughter nuclei. Shaded area corresponds to the maximum probability for finding the mother and granddaughter states. (b) Pairing energies as a function of the intrinsic deformation.

Figure 3

Nuclear matrix elements calculated for different methods (ISM [5, 22], QRPA(Jy) [8], QRPA(Tu) [7], IBM-2 [12], PHFB [10]) with UCOM short-range correlations. QRPA values are calculated with $g_A=1.25$ and IBM-2 and PHFB results are multiplied by 1.18 to account for the difference between Jastrow and UCOM [29].