Improved description of the $2\nu \beta \beta$-decay and a possibility to determine the effective axial-vector coupling constant

An improved formalism of the two-neutrino double-beta decay ($2\nu \beta \beta$-decay) rate is presented, which takes into account the dependence of energy denominators on lepton energies via the Taylor expansion. Until now, only the leading term in this expansion has been considered. The revised $2\nu \beta \beta$-decay rate and differential characteristics depend on additional phase-space factors weighted by the ratios of $2\nu \beta \beta$-decay nuclear matrix elements with different powers of the energy denominator. For nuclei of experimental interest all phase-space factors are calculated by using exact Dirac wave functions with finite nuclear size and electron screening. For isotopes with measured $2\nu \beta \beta$-decay half-life the involved nuclear matrix elements are determined within the quasiparticle random-phase approximation with partial isospin restoration. The importance of correction terms to the $2\nu \beta \beta$-decay rate due to Taylor expansion is established and the modification of shape of single and summed electron energy distributions is discussed. It is found that the improved calculation of the $2\nu \beta \beta$-decay predicts slightly suppressed $2\nu \beta \beta$-decay background to the neutrinoless double-beta decay signal. Furthermore, an approach to determine the value of effective weak-coupling constant in nuclear medium $g_{\mathrm{A}}^{\mathrm{eff}}$ is proposed.

Figure 1

Running sum of the $2\nu \beta \beta$-decay NMEs $M_{GT-1}^{2\nu }$ and $M_{GT-3}^{2\nu }$ [see Eq. (17)] for ${}^{48}\mathrm{Ca}$, ${}^{116}\mathrm{Cd}$, ${}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$ (normalized to unity) as a function of the excitation energy $E_{\mathrm{ex}}$ counted from the ground state of intermediate nucleus. Calculations were performed within the proton-neutron QRPA with isospin restoration [24]. Results are obtained with Argonne V18 potential and for unquenched axial-vector coupling constant $g_A=1.269$.

Figure 2

The same as Fig. 1 but for the $2\nu \beta \beta$-decay of ${}^{76}\mathrm{Ge}$, ${}^{82}\mathrm{Se}$, ${}^{96}\mathrm{Zr}$, and ${}^{100}\mathrm{Mo}$.

Figure 3

Partial differential decay rates $(1/{\mathrm{\Gamma }}_0)d{\mathrm{\Gamma }}_0/d{T}_e$, $(1/{\mathrm{\Gamma }}_2)d{\mathrm{\Gamma }}_2/d{T}_e$, and $(1/{\mathrm{\Gamma }}_4)d{\mathrm{\Gamma }}_4/d{T}_e$ normalized to corresponding partial decay rate vs kinetic energy of a single electron $T_e$ (in units of $Q$ value) (left panels) and the partial differential decay rates $(1/{\mathrm{\Gamma }}_0)d{\mathrm{\Gamma }}_0/d{T}_{ee}$, $(1/{\mathrm{\Gamma }}_2)d{\mathrm{\Gamma }}_2/d{T}_{ee}$, and $(1/{\mathrm{\Gamma }}_4)d{\mathrm{\Gamma }}_4/d{T}_{ee}$ normalized to corresponding partial decay rate vs the sum of kinetic energies of emitted electrons $T_{ee}$ (in units of $Q$ value) (right panels) for the $2\nu \beta \beta$-decay of ${}^{82}\mathrm{Se}$ and ${}^{100}\mathrm{Mo}$ to the ground state of the final nucleus. The energy distributions are normalized to unity to see how their shapes differ.

Figure 4

The full differential decay rate $(1/\mathrm{\Gamma })d\mathrm{\Gamma }/d{T}_e$ and partial differential decay rates $(1/\mathrm{\Gamma })d{\mathrm{\Gamma }}_0/d{T}_e$ and $(1/\mathrm{\Gamma })d{\mathrm{\Gamma }}_2/d{T}_e$ normalized by the full decay rate vs the kinetic energy of a single electron $T_e$ (in units of $Q$ value) for the $2\nu \beta \beta$-decay of ${}^{48}\mathrm{Ca}$, ${}^{116}\mathrm{Cd}$, ${}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$.

Figure 5

The same as Fig. 4 but for the $2\nu \beta \beta$-decay of ${}^{76}\mathrm{Ge}$, ${}^{82}\mathrm{Se}$, ${}^{96}\mathrm{Zr}$, and ${}^{100}\mathrm{Mo}$.

Figure 6

The full differential decay rate $(1/\mathrm{\Gamma })d\mathrm{\Gamma }/d{T}_e$ and partial differential decay rates $(1/\mathrm{\Gamma })d{\mathrm{\Gamma }}_0/d{T}_e$ and $(1/\mathrm{\Gamma })d{\mathrm{\Gamma }}_2/d{T}_e$ normalized by the full decay rate vs the sum of kinetic energies of emitted electrons $T_{ee}$ (in units of $Q$ value) for the $2\nu \beta \beta$-decay of ${}^{48}\mathrm{Ca}$, ${}^{116}\mathrm{Cd}$, ${}^{130}\mathrm{Te}$, and ${}^{136}\mathrm{Xe}$.

Figure 7

The same as Fig. 6 for the $2\nu \beta \beta$-decay of ${}^{76}\mathrm{Ge}$, ${}^{82}\mathrm{Se}$, ${}^{96}\mathrm{Zr}$, and ${}^{100}\mathrm{Mo}$.

Figure 8

Differential decay rates $(1/\mathrm{\Gamma })d\mathrm{\Gamma }/d(T_e/Q)$ (upper panels) and $(1/\mathrm{\Gamma })d\mathrm{\Gamma }/d(T_{ee}/Q)$ (lower panels) normalized by the full decay rate $\mathrm{\Gamma }$ vs kinetic energy of a single electron $T=T_e$ and the sum of kinetic energies of emitted electrons $T=T_{ee}$ (in units of $Q$ value), respectively. Results are presented for the $2\nu \beta \beta$-decay of ${}^{82}\mathrm{Se}$ (left panels) and ${}^{100}\mathrm{Mo}$ (right panels) by assuming ${\xi }_{13}^{2\nu }=0.0$, 0.40, and 0.8.

Figure 9

The endpoint of the spectrum of the differential decay rate normalized to the full decay rate $(1/\mathrm{\Gamma })d\mathrm{\Gamma }/dT$ vs the sum of kinetic energy of emitted electrons $T=(E_{e_1}+E_{e_2}-2m_e)$ for the $2\nu \beta \beta$-decay of ${}^{82}\mathrm{Se}$ and ${}^{100}\mathrm{Mo}$. The calculation with the standard (leading term in Taylor expansion) and improved (present work) theoretical description of the $2\nu \beta \beta$-decay rate. The considered ratios ${\xi }_{31}^{2\nu }$ and ${\xi }_{51}^{2\nu }$ are those calculated within the QRPA with isospin restoration (see Table 2).

Figure 10

The ratio ${\xi }_{31}^{2\nu }$ of nuclear matrix elements $M_{GT-3}^{2\nu }$ and $M_{GT-1}^{2\nu }$ calculated within the $pn\mathrm{QRPA}$ with partial restoration of isospin symmetry [24] by assuming $g_A^{\mathrm{eff}}=0.80$, 1.00, 1nd 1.269, the single-state dominance hypothesis (SSD) [13, 14], and by using the Gamow–Teller strengths measured in charge-exchange reactions (CheR) [29, 30, 31, 32] under the assumption of equal phases of all contributions to the matrix element.

Figure 11

The effective axial-vector coupling constant $g_A^{\mathrm{eff}}$ as function of the matrix element $M_{GT-3}^{2\nu }$ for $2\nu \beta \beta$-decay of ${}^{100}\mathrm{Mo}$ and ${}^{116}\mathrm{Cd}$. The SSD values are assumed for ${\xi }_{31}^{2\nu }$ (see Table 3).