Octupole correlations in low-lying states of ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$ and their impact on neutrinoless double-$\beta$ decay

We present a generator-coordinate calculation, based on a relativistic energy-density functional, of the low-lying spectra in the isotopes ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$ and of the nuclear matrix element that governs the neutrinoless double-$\beta$ decay of the first isotope to the second. We carefully examine the impact of octupole correlations on both nuclear structure and the double-$\beta$ decay matrix element. Octupole correlations turn out to reduce quadrupole collectivity in both nuclei. Shape fluctuations, however, dilute the effects of octupole deformation on the double-$\beta$ decay matrix element, so that the overall octupole-induced quenching is only about $7\%$.

Figure 1

Mean-field energy surfaces for ${}^{150}\mathrm{Nd}$ (a) and ${}^{150}\mathrm{Sm}$ (b), projected energy surfaces for ${}^{150}\mathrm{Nd}$ (c) and ${}^{150}\mathrm{Sm}$ (d), and the square of the collective ground-state wave function for ${}^{150}\mathrm{Nd}$ (e) and ${}^{150}\mathrm{Sm}$ (f), all in the ${\beta }_2-{\beta }_3$ plane.

Figure 2

Low-energy spectra of ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$. The numbers on arrows are $E2$ (solid line) and $E3$ (dashed line) transition strengths, in Weisskopf units. Data are from Ref. [68].

Figure 3

Excitation energies of parity doublet states in ${}^{150}\mathrm{Nd}$ (a) and ${}^{150}\mathrm{Sm}$ (b). The available data ($\blacksquare$) are compared with the GCM results ($\square$) and the results produced by the single-configuration of $J=0$ energy minimum ($▵$) and by the configuration with deformation parameters determined by measured transition strengths $B(E2:0_1^{+}\to 2_1^{+})$ and $B(E3:0_1^{+}\to 3_1^{-})(▿)$ [68].

Figure 4

Normalized nuclear matrix elements ${\tilde{M}}^{0\nu }\left(q_I,q_F\right)$ for the neutrinoless double-$\beta$ decay of ${}^{150}\mathrm{Nd}$, where $\left\{q\right\}\equiv \{{\beta }_2,{\beta }_3\}$. (a) plots ${\tilde{M}}^{0\nu }$ versus the initial and final octupole deformation parameters, with the quadruple deformation parameters ${\beta }_2^I$ and ${\beta }_2^F$ fixed at 0.2. (b) plots the same quantity with the restriction ${\beta }_3^I={\beta }_3^F$.

Figure 5

The final matrix element $M^{0\nu }$ from the GCM calculation with and without [46] octupole shape fluctuations (REDF) and those of the QRPA (“QRPA_F” [70], “QRPA_M” [45], “QRPA_T” [47]), the IMB-2 [71], and the nonrelativistic GCM, based on the Gogny D1S interaction, with [72] and without [44] pairing fluctuations.