Figure 1

Double-$\beta$ decay mechanism for (a) two-neutrino, (b) neutrinoless, and (c) neutrinoless decay with Majoron emission.Reuse & Permissions

Figure 2

Electron radial wave functions $g_{-1}(\epsilon ,r)$, $f_{-1}(\epsilon ,r)$ (left panel) and $f_1(\epsilon ,r)$, $g_1(\epsilon ,r)$ (right panel) for $Z_d=62$, $\epsilon =2.0$ MeV, and $R=6.38$ fm (vertical line). The notations WF1, WF2, and WF3 correspond to leading finite-size Coulomb, exact finite-size Coulomb, and exact finite-size Coulomb with electron screening, respectively.Reuse & Permissions

Figure 3

Notation used in this paper. The example is for ${}^{150}$Nd decay.Reuse & Permissions

Figure 4

(a) Skeleton of the ${}^{110}$Pd decay scheme. The ground state of the intermediate ${}^{110}$Ag nucleus is $1^{+}$, leading to the lowest possible value for $E_N$ to be $E_{1_1^{+}}=0.0$ MeV. (b) Behavior of the phase-phase factor $G_{2\nu }^{\left(0\right)}$ as a function of $\tilde{A}$. The value obtained using the single-state dominance hypothesis, $\tilde{A}=1.893$ MeV, is denoted by a red circle and the value obtained using $\tilde{A}=1.12\times {110}^{1/2}$ MeV $=11.75$ MeV is denoted by a blue square.Reuse & Permissions

Figure 5

Phase-space factors $G_{2\nu }^{\left(0\right)}$ in units of ${10}^{-21}$ yr${}^{-1}$. The label “approximate” refers to the results obtained by the use of approximate electron wave functions. The figure is in semilogarithmic scale.Reuse & Permissions

Figure 6

Single-electron spectra (left panel), summed energy spectra (middle panel), and angular correlations between two outgoing electrons (right panel) for the ${}^{136}$Xe ${\to }^{136}$Ba $2\nu \beta \beta$ decay. The scale in the left and middle panels should be multiplied by ${\mathcal{N}}_{2\nu }$ when comparing with experiment.Reuse & Permissions

Figure 7

Same as Fig. 6 but for the ${}^{82}$Se ${\to }^{82}$Kr $2\nu \beta \beta$ decay.Reuse & Permissions

Figure 8

Same as Fig. 6 but for the ${}^{150}$Nd ${\to }^{150}$Sm $2\nu \beta \beta$ decay. The figure also shows the difference between our “exact” calculation and the previously used approximate calculation.Reuse & Permissions

Figure 9

Single-electron spectra for the ${}^{110}$Pd ${\to }^{110}$Cd $2\nu \beta \beta$ decay obtained using the two approximations discussed in the text, namely, the closure approximation and the single-state dominance hypothesis. The scale should be multiplied by ${\mathcal{N}}_{2\nu }$ when comparing with experiment.Reuse & Permissions

Figure 10

Phase-space factors $G_{0\nu }^{\left(0\right)}$ in units of ${10}^{-15}$ yr${}^{-1}$. The label “approximate” refers to the results obtained by the use of approximate electron wave functions. The figure is in semilogarithmic scale.Reuse & Permissions

Figure 11

Single-electron spectra (left panel) and angular correlations between the two outgoing electrons (right panel) for the ${}^{76}$Ge ${\to }^{76}$Se $0\nu \beta \beta$ decay. The scale of the left panel should be multiplied by ${\mathcal{N}}_{0\nu }$ for a realistic estimate.Reuse & Permissions

Figure 12

Effective nuclear matrix elements $|M_{2\nu }^{\mathrm{eff}}|$ extracted from the experimental $2\nu \beta \beta$ half-lives as a function of mass number.Reuse & Permissions