Bound-state double-$\beta$ decay

We consider alternative modes of two-neutrino and neutrinoless double-$\beta$ decays in which one $\beta$ electron goes over to a continuous spectrum and the other occupies a vacant bound level of the daughter ion. We calculate the corresponding phase-space factors of the final states, estimate the partial decay rates, and derive the one- and two-electron energy spectra using relativistic many-electron wave functions of atoms provided by the multiconfiguration Dirac-Hartree-Fock package Grasp2K. While the bound-state neutrinoless double-$\beta$ decays are strongly suppressed, their two-neutrino counterparts can be observed in the next-generation double-$\beta$-decay experiments, most notably SuperNEMO.

Figure 1

A schematic view of the $0\nu \left(2\nu \right)\mathrm{EP}\beta$ decays. The final state involves the daughter nucleus ${}_{Z+2}{}^A\mathrm{Y}$, a bound electron $e_{\mathrm{b}}^{-}$ produced above the subshells occupied by $Z$ atomic electrons, and a single free electron $e^{-}$ (and a pair of electron antineutrinos ${\overline{\nu }}_e$) emitted from the atom. Upon the deexcitation, the bound electron $e_{\mathrm{b}}^{-}$ radiates photons of energy $\lesssim 10\mathrm{eV}$.

Figure 2

The squared radial wave functions $f_{n,-1}^2\left(R\right)$ and $g_{n,+1}^2\left(R\right)$ (in atomic units) for the subshells $8{\mathrm{s}}_{1/2}$ and $8{\mathrm{p}}_{1/2}$, respectively, fitted by the power function $a{Z}^b$ of the initial atomic number $Z$. The points represent the predictions of Grasp2K. The parameters determined from the fit read as follows: $a=1.1\times {10}^{-10}, b=6.2$ ($8{\mathrm{s}}_{1/2}$) and $a=8.1\times {10}^{-12}, b=6.4$ ($8{\mathrm{p}}_{1/2}$).

Figure 3

The squared radial wave functions $f_{n,-1}^2\left(R\right)$ and $g_{n,+1}^2\left(R\right)$ (in atomic units) for the isotope ${}^{82}\mathrm{Se}$ fitted by the power function $c{n}^d$ of the principal quantum number $n$. The parameters determined from the fit read as follows: $c=1.1\times {10}^6, d=-6.1$ ($n{\mathrm{s}}_{1/2}$) and $c=4.6\times {10}^6, d=-10$ ($n{\mathrm{p}}_{1/2}$).

Figure 4

The Fermi sum ${\sum }_{n=n_{\min }}^{\infty }{B}_n(Z+2)$ (in atomic units) as a function of the initial atomic number $Z$ of the parent nucleus for $n_{\min }=5,6,7$.

Figure 5

The decay-rate ratio ${\mathrm{\Gamma }}^{0\nu \mathrm{EP}\beta }/{\mathrm{\Gamma }}^{0\nu \beta \beta }$ as a function of the atomic number $Z$ of the parent nucleus and the $Q$ value. The $0\nu \mathrm{EP}\beta$ decay rate is maximal for the isotopes: ${}^{98}\mathrm{Mo}, {}^{80}\mathrm{Se}$, and ${}^{146}\mathrm{Nd}$, and decreases rapidly with increasing both $Z$ and $Q$.

Figure 6

The decay-rate ratio ${\mathrm{\Gamma }}^{2\nu \mathrm{EP}\beta }/{\mathrm{\Gamma }}^{2\nu \beta \beta }$ as a function of the atomic number $Z$ of the parent nucleus and the $Q$ value. The two-neutrino mode exhibits behavior similar to the neutrinoless mode (see Fig. 5), but the absolute values are by one order of magnitude higher.

Figure 7

The half-lives $T_{1/2}^{0\nu \mathrm{EP}\beta }$ and $T_{1/2}^{0\nu \beta \beta }$ for the isotopes with the calculated NMEs [23, 24] estimated assuming the unquenched axial coupling constant: $g_A=1.27$ and the effective Majorana neutrino mass: $\left|m_{\beta \beta }\right|=50\mathrm{meV}$.

Figure 8

The half-lives $T_{1/2}^{2\nu \mathrm{EP}\beta }$ and $T_{1/2}^{2\nu \beta \beta }$ [25] for the isotopes with double-$\beta$ decays observed experimentally. The $2\nu \mathrm{EP}\beta$ and $0\nu \beta \beta$ decay rates are comparable in magnitude.

Figure 9

The $0\nu \mathrm{EP}\beta$ and $0\nu \beta \beta$ one-electron spectra $(1/{\mathrm{\Gamma }}^{0\nu \beta \beta })d\mathrm{\Gamma }/d\varepsilon$ as functions of the normalized electron kinetic energy $\varepsilon =(E-m_e)/Q$ for the isotope ${}^{82}\mathrm{Se}$. The $0\nu \mathrm{EP}\beta$ peak is represented by a Gaussian with $\mathrm{FWHM}/Q=7\%/\sqrt{Q/\mathrm{MeV}}$, which corresponds to the planned energy resolution of the SuperNEMO calorimeters, and scaled by a factor of ${10}^4$. The composition of the $0\nu \mathrm{EP}\beta$ peak beyond the endpoint $\varepsilon =1$ is shown in the upper left corner.

Figure 10

The $2\nu \mathrm{EP}\beta$ and $2\nu \beta \beta$ one-electron spectra $(1/\mathrm{\Gamma })d\mathrm{\Gamma }/d\varepsilon$ as functions of the normalized electron kinetic energy $\varepsilon =(E-m_e)/Q$ for ${}^{82}\mathrm{Se}$.

Figure 11

The $2\nu \mathrm{EP}\beta$ and $2\nu \beta \beta$ normalized differential decay rates $(1/\mathrm{\Gamma })d\mathrm{\Gamma }/d{\varepsilon }_{12}$ as functions of the sum ${\varepsilon }_{12}={\varepsilon }_1+{\varepsilon }_2$ of electron energies for ${}^{76}\mathrm{Ge}$.

Figure 12

The ratios (27) between the integrated $2\nu \mathrm{EP}\beta$ and $2\nu \beta \beta$ decay rates as functions of the energy intervals $[{\varepsilon }_{\min },{\varepsilon }_{\max }]$ and $[{\varepsilon }_{12\min },{\varepsilon }_{12\max }]$. The one-electron ratio is given for ${}^{82}\mathrm{Se}$ (left panel) and the two-electron ratio refers to ${}^{76}\mathrm{Ge}$ (right panel). The ROIs with the highest $2\nu \mathrm{EP}\beta$ sensitivity belong to the opposite sides of the ${\varepsilon }_{\left(12\right)}$-interval at $\varepsilon =1$ and ${\varepsilon }_{12}=0$, respectively.