Relativistic description of nuclear matrix elements in neutrinoless double-$\beta$ decay

Background: Neutrinoless double-$\beta \left(0\nu \beta \beta \right)$ decay is related to many fundamental concepts in nuclear and particle physics beyond the standard model. Currently there are many experiments searching for this weak process. An accurate knowledge of the nuclear matrix element for the $0\nu \beta \beta$ decay is essential for determining the effective neutrino mass once this process is eventually measured.

Purpose: We report the first full relativistic description of the $0\nu \beta \beta$ decay matrix element based on a state-of-the-art nuclear structure model.

Methods: We adopt the full relativistic transition operators which are derived with the charge-changing nucleonic currents composed of the vector coupling, axial-vector coupling, pseudoscalar coupling, and weak-magnetism coupling terms. The wave functions for the initial and final nuclei are determined by the multireference covariant density functional theory (MR-CDFT) based on the point-coupling functional PC-PK1. Correlations beyond the mean field are introduced by configuration mixing of both angular momentum and particle number projected quadrupole deformed mean-field wave functions.

Results: The low-energy spectra and electric quadrupole transitions in ${}^{150}\mathrm{Nd}$ and its daughter nucleus ${}^{150}\mathrm{Sm}$ are well reproduced by the MR-CDFT calculations. The $0\nu \beta \beta$ decay matrix elements for both the $0_1^{+}\to 0_1^{+}$ and $0_1^{+}\to 0_2^{+}$ decays of ${}^{150}\mathrm{Nd}$ are evaluated. The effects of particle number projection, static and dynamic deformations, and the full relativistic structure of the transition operators on the matrix elements are studied in detail.

Conclusions: The resulting $0\nu \beta \beta$ decay matrix element for the $0_1^{+}\to 0_1^{+}$ transition is $5.60$, which gives the most optimistic prediction for the next generation of experiments searching for the $0\nu \beta \beta$ decay in ${}^{150}\mathrm{Nd}$.

Figure 1

Average pairing gap ${\Delta }^{v^2}$ for (a) neutrons and (b) protons in ${}^{150}\mathrm{Nd}$ as a function of deformation $\beta$ obtained by the $\mathrm{RMF}+\mathrm{BCS}$ method, using the separable finite-range pairing force (BCS-s) and the $\delta$ pairing force with adjusted strength constants $V_{\tau }$ (BCS-$\delta$), respectively.

Figure 2

Single-configuration matrix element ${\tilde{M}}^{0\nu }({\beta }_I,{\beta }_F)$ defined in Eq. (24) between the spherical states of ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$, as a function of the cutoff parameters ${\zeta }_1$ and ${\zeta }_2$, respectively. The horizontal dash-dotted line denotes the value corresponding to ${\zeta }_1=0$ and ${\zeta }_2={10}^{-5}$.

Figure 3

Single-configuration matrix element ${\tilde{M}}^{0\nu }({\beta }_I,{\beta }_F)$ between the spherical states of ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$, as a function of the energy denominator $E_d$ in Eq. (10). The empirical value of $E_d=1.12A^{1/2}\mathrm{MeV}$ is marked by a vertical dash-dotted line.

Figure 4

The intrinsic (RMF) and the PNAMP $\left(N\&Z,J=0,2,4,6\right)$ PECs, together with the energy and the average axial deformation of the lowest GCM state for each angular momentum in ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$. The AMP PECs, which are provided by calculations without exact number projection, are also presented for $J=0$.

Figure 5

Squares of collective wave functions $|g_{\alpha }^{J=0}{\left(\beta \right)|}^2$ obtained by the $\mathrm{GCM}+\mathrm{PNAMP}$ and $\mathrm{GCM}+\mathrm{AMP}$ methods for the ground states of ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$, as well as for the first excited $0^{+}$ state of ${}^{150}\mathrm{Sm}$.

Figure 6

Low-lying energy levels and $E2$ transition probabilities for the nuclei ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$ obtained by the $\mathrm{GCM}+\mathrm{PNAMP}$ and $\mathrm{GCM}+\mathrm{AMP}$ methods in comparison with experimental data.

Figure 7

Single-configuration matrix elements ${\tilde{M}}^{0\nu }({\beta }_I,{\beta }_F)$ defined in Eq. (24) with ${\beta }_I={\beta }_F$ for transitions from ${}^{150}\mathrm{Nd}$ to ${}^{150}\mathrm{Sm}$, obtained by calculations with PNP (PNAMP) and without (AMP).

Figure 8

(a) Distributions of the total transition matrix element $M^{0\nu }$ of Eq. (21) between the ground states of ${}^{150}\mathrm{Nd}$ and ${}^{150}\mathrm{Sm}$ in the various regions of the ${\beta }_I\text{−}{\beta }_F$ plane, calculated with the $\mathrm{GCM}+\mathrm{PNAMP}$ method and (c) normalized matrix element ${\tilde{M}}^{0\nu }({\beta }_I,{\beta }_F)$ of Eq. (24) obtained by the single-configuration calculation with PNAMP. (b) Squares of ground-state wave functions obtained with the $\mathrm{GCM}+\mathrm{PNAMP}$ method and (d) pairing energies (33) from the $\mathrm{RMF}+\mathrm{BCS}$ calculation for initial and final nuclei are shown for comparison.

Figure 9

Contribution from each coupling channel to the total NMEs of the $0\nu \beta \beta$ decay from ${}^{150}\mathrm{Nd}$ to ${}^{150}\mathrm{Sm}$ for both the (a) $0_1^{+}\to 0_1^{+}$ and the (b) $0_1^{+}\to 0_2^{+}$ transitions. Values of the matrix elements evaluated using the full relativistic operator $M^{0\nu }$ (Rel.) are compared with those obtained with the nonrelativistic reduced operator $M_{\mathrm{NR}}^{0\nu }$ (Non-rel.). The results are calculated with the $\mathrm{GCM}+\mathrm{PNAMP}$ method.