Neutrinoless $\beta \beta$ decays to excited $0^{+}$ states and the Majorana-neutrino mass

The nuclear matrix elements (NMEs) corresponding to the neutrinoless double-$\beta$ ($0\nu \beta \beta$) decays to excited $0^{+}$ states of major experimental interest are calculated. All these decay transitions are electron emitting ($0\nu {\beta }^{-}{\beta }^{-}$ decays) and take place in the mass $A=76,82,96,100,110,116,124,130,136$ nuclei. This work is an extension of our previous work [Phys. Rev. C 91, 024613 (2015)], where $0\nu \beta \beta$ decays to the ground states of the same nuclei were treated. We calculate the NMEs for transitions mediated by both the light (l-NMEs) and the heavy (h-NMEs) Majorana neutrinos. A higher-QRPA (quasiparticle random-phase approximation) framework, the multiple-commutator model, is adopted for the calculations, including a previously omitted contribution to the transitions to two-phonon states. A Bonn $G$-matrix-based effective nucleon-nucleon interaction is generated by exploiting the recently proposed isoscalar-isovector decomposition of the particle-particle proton-neutron interaction parameter, $g_{\mathrm{pp}}$. All the appropriate short-range correlations, nucleon form factors, and higher-order nucleonic weak currents are included to benchmark our calculations. The relevant nuclear spectroscopy was checked to validate the nuclear models used. The computed l-NMEs and h-NMEs are compared with the available other calculations and the relevance of the new included two-phonon term is discussed. The results are summarized by easy-to-use half-life-Majorana-mass interrelations.

Figure 1

Theoretical and experimental low-energy $0^{+}$ spectra for the nuclei ${}^{96}\mathrm{Mo}$ and ${}^{116}\mathrm{Sn}$. The experimental information is found from Ref. [36] (${}^{96}\mathrm{Mo}$) and Ref. [37] (${}^{116}\mathrm{Sn}$).

Figure 2

Theoretical and experimental low-energy $2^{+}$ spectra for the nuclei ${}^{96}\mathrm{Mo}$ and ${}^{116}\mathrm{Sn}$. The experimental information is found from Ref. [36] (${}^{96}\mathrm{Mo}$) and Ref. [37] (${}^{116}\mathrm{Sn}$).

Figure 3

Theoretical and experimental low-energy $2^{+}$ spectra for the nuclei ${}^{82}\mathrm{Kr}$ and ${}^{124}\mathrm{Te}$. The experimental information is found from Ref. [38] (${}^{82}\mathrm{Kr}$) and Ref. [39] (${}^{124}\mathrm{Te}$).

Figure 4

Theoretical and experimental low-energy $0^{+}$ and $2^{+}$ spectra for the nuclei ${}^{82}\mathrm{Kr}$ and ${}^{130}\mathrm{Xe}$, respectively. The experimental information is found from Ref. [38] (${}^{82}\mathrm{Kr}$) and Ref. [40] (${}^{130}\mathrm{Xe}$).

Figure 5

Theoretical and experimental low-energy $0^{+}$ and $2^{+}$ spectra for the nucleus ${}^{136}\mathrm{Ba}$. The experimental information is found from Ref. [41].

Figure 6

Multipole decomposition of the l-NME $M^{\left(0\nu \right)}$ for nuclei ${}^{76}\mathrm{Ge}, {}^{82}\mathrm{Se}, {}^{96}\mathrm{Zr}, {}^{100}\mathrm{Mo}, {}^{110}\mathrm{Pd}, {}^{116}\mathrm{Cd}, {}^{124}\mathrm{Sn}, {}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$ for transitions to the first excited state in the corresponding daughter nuclei.

Figure 7

Effect of the previously omitted second term of the two-phonon transition density (20) on the total l-NMEs. Dark (light) gray bars give the matrix element without (with) the second term (20) included. The lowest row of numbers displays the shifts of the NMEs in percents.

Figure 8

Pair-angular-momentum decomposition of the l-NME for nuclei ${}^{76}\mathrm{Ge},{}^{82}\mathrm{Se},{}^{100}\mathrm{Mo}$, and ${}^{136}\mathrm{Xe}$. Dark (light) gray bars show the corresponding $J^{\prime }$ contribution without (with) the second term (20) of the two-phonon transition density.

Figure 9

Pair-angular-momentum decomposition of the l-NME for nuclei ${}^{96}\mathrm{Zr}$ and ${}^{116}\mathrm{Cd}$.

Figure 10

Radial dependence of the l-NMEs $M_{\mathrm{F}}^{\left(0\nu \right)}, M_{\mathrm{GT}}^{\left(0\nu \right)}$, and the full matrix element $M^{\left(0\nu \right)}$ for nuclei ${}^{76}\mathrm{Ge}$ and ${}^{96}\mathrm{Zr}$. These NMEs correspond to the $0_{\mathrm{gs}}⟶0_1^{+}$ transitions.

Figure 11

Effect of the previously omitted second term of the two-phonon transition density (20) on the total h-NMEs. Dark (light) gray bars give the matrix element without (with) the second term (20) included. The lowest row of numbers displays the shifts of the NMEs in percents. These NMEs are computed with the CD-Bonn parametrization of the SRC effects and with $g_{\mathrm{A}}=1.00$.

Figure 12

Pair-angular-momentum decomposition of the h-NME for nuclei ${}^{76}\mathrm{Ge},{}^{82}\mathrm{Se},{}^{110}\mathrm{Pd}$, and ${}^{136}\mathrm{Xe}$. Dark (light) gray bars show the corresponding $J^{\prime }$ contribution without (with) the second term (20) of the two-phonon transition density.

Figure 13

Multipole decomposition of the h-NME $M^{\left(0\nu \right)}$ for nuclei ${}^{82}\mathrm{Se}, {}^{100}\mathrm{Mo}, {}^{110}\mathrm{Pd}$, and ${}^{116}\mathrm{Cd}$ for transitions to the first $0^{+}$ excited state in the corresponding daughter nuclei.

Figure 14

Multipole decomposition of the h-NME $M^{\left(0\nu \right)}$ for nuclei ${}^{76}\mathrm{Ge}, {}^{96}\mathrm{Zr}, {}^{124}\mathrm{Sn},{}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$ for transitions to the first $0^{+}$ excited state in the corresponding daughter nuclei.