Structure of the two-neutrino double-$\beta$ decay matrix elements within perturbation theory

The two-neutrino double-$\beta$ Gamow-Teller and Fermi transitions are studied within an exactly solvable model, which allows a violation of both spin-isospin SU(4) and isospin SU(2) symmetries, and is expressed with generators of the SO(8) group. It is found that this model reproduces the main features of realistic calculation within the quasiparticle random-phase approximation with isospin symmetry restoration concerning the dependence of the two-neutrino double-$\beta$ decay matrix elements on isovector and isoscalar particle-particle interactions. By using perturbation theory an explicit dependence of the two-neutrino double-$\beta$ decay matrix elements on the like-nucleon pairing, particle-particle $T=0$ and $T=1$, and particle-hole proton-neutron interactions is obtained. It is found that double-$\beta$ decay matrix elements do not depend on the mean field part of Hamiltonian and that they are governed by a weak violation of both SU(2) and SU(4) symmetries by the particle-particle interaction of Hamiltonian. It is pointed out that there is a dominance of two-neutrino double-$\beta$ decay transition through a single state of intermediate nucleus. The energy position of this state relative to energies of initial and final ground states is given by a combination of strengths of residual interactions. Further, energy-weighted Fermi and Gamow-Teller sum rules connecting $\mathrm{\Delta }Z=2$ nuclei are discussed. It is proposed that these sum rules can be used to study the residual interactions of the nuclear Hamiltonian, which are relevant for charge-changing nuclear transitions.

Figure 1

Eigenenergies $E_{STn}$ of the Hamiltonian (13) for set of parameters (16), $T_z=4$, 3, and 2 and by assuming $H_I=0$. Energy states are labeled by spin, isospin, and the SU(4) quantum number $n$: ($S,T,n$). The levels with different value of $S+T$ are displayed in different colors online: $S+T=2$ (green), 4 (red), 6 (orange), 8 (black), and 10 (blue).

Figure 2

Matrix element $M_{GT}^{2\nu }$ for the double-GT two-neutrino double-$\beta$ decay mode as function of ratio $g_{pp}^{T=0}/g_{\mathrm{pair}}$ for a set of parameters (16). Exact results are indicated with a solid line. The results obtained within the perturbation theory up to the first and second order in $H_I$ contribution to Hamiltonian (13) are shown with dash-dotted and dashed lines, respectively. The restoration of spin-isospin symmetry of particle-particle interaction is achieved for $g_{pp}^{T=0}/g_{\mathrm{pair}}=1$.

Figure 3

Matrix element $M_{GT}^{2\nu }$ for the double-GT two-neutrino double-$\beta$ decay mode as function of ratio $g_{pp}^{T=0}/g_{\mathrm{pair}}$ for different initial state with $T=M_T$ ($M_T=2$, 4, 6, 8, and 10).

Figure 4

Matrix elements $M_F^{2\nu }$ and $M_{GT}^{2\nu }$ as function of ratios $g_{pp}^{T=0}/g_{\mathrm{pair}}$ and $g_{pp}^{T=1}/g_{\mathrm{pair}}$ for transition from the initial $|{04}^{\prime }{4}^{\prime }\rangle$ to final $|{02}^{\prime }{2}^{\prime }\rangle$ ground state and a set of parameters (16). The results are obtained within the perturbation theory up to the second order.