$0\nu \beta \beta$ and $2\nu \beta \beta$ nuclear matrix elements evaluated in closure approximation, neutrino potentials and SU(4) symmetry

The intimate relation between the Gamow-Teller part of the matrix element $M_{GT}^{0\nu }$ and the $2\nu \beta \beta$ closure matrix element $M_{cl}^{2\nu }$ is explained and explored. If the corresponding radial dependence $C_{cl}^{2\nu }\left(r\right)$ would be known, $M^{0\nu }$ corresponding to any mechanism responsible for the $0\nu \beta \beta$ decay can be obtained as a simple integral. However, the $M_{cl}^{2\nu }$ values, and therefore also the functions $C_{cl}^{2\nu }\left(r\right)$, sensitively depend not only on the properties of the first few $1^{+}$ states but also of higher-lying $1^{+}$ states in the intermediate odd-odd nuclei. We show that the ${\beta }^{-}$ and ${\beta }^{+}$ amplitudes of such states typically have opposite relative signs, and their contributions reduce severally the $M_{cl}^{2\nu }$ values. We suggest that demanding that $M_{cl}^{2\nu }=0$ is a sensible alternative way, within the QRPA method, of determining the amount of renormalization of isoscalar particle-particle interaction strength $g_{pp}^{T=0}$. Using such prescription, the matrix elements $M^{0\nu }$ are evaluated; their values are not very different $\left(\le 20\%\right)$ from the usual QRPA values when $g_{pp}^{T=0}$ is related to the known $2\nu \beta \beta$ half-lives. We note that vanishing values of $M_{cl}^{2\nu }$ are signs of a partial restoration of the spin-isospin SU(4) symmetry.

Figure 1

Functions $C_{GT}^{0\nu }\left(r\right)$ evaluated in the QRPA for a number of $0\nu \beta \beta$ candidate nuclei.

Figure 2

Functions $C_{GT}^{0\nu }\left(r\right)$ evaluated in the QRPA for several Ca isotopes. ${}^{48}\mathrm{Ca}$ is a real $\beta \beta$ decay candidate. It decays into ${}^{48}\mathrm{Ti}$ and the isospin $T$ changes in the decay by two units $\left(\mathrm{\Delta }T=2\right)$. The other two Ca isotopes cannot $\beta \beta$ decay; nevertheless the corresponding matrix elements can be evaluated. The transition ${}^{42}\mathrm{Ca}\to {}^{42}\mathrm{Ti}$ connects mirror nuclei, the isospin does not change, $\mathrm{\Delta }T=0$.

Figure 3

Functions $C_I^{0\nu }\left(r\right)$ evaluated in the QRPA for a the $I$ = Gamow-Teller, Fermi, and tensor matrix elements for ${}^{48}\mathrm{Ca}0\nu \beta \beta$ decay.

Figure 4

Functions $C\left(r\right)$ corresponding to the total strengths $S\left({\beta }^{-}\right)$ and $S\left({\beta }^{+}\right)$ for the initial nucleus ${}^{76}\mathrm{Ge}$ and for the final nucleus ${}^{76}\mathrm{Se}$.

Figure 5

Cumulative contributions to the $M^{2\nu }$ (11) and $M_{GTcl}^{2\nu }$ (12) as a function of the intermediate state excitation energy. This is for the case of ${}^{76}\mathrm{Ge}$.

Figure 6

Functions $C_{cl}^{2\nu }\left(r\right)$ for several $\beta \beta$ candidate nuclei evaluated within the QRPA.

Figure 7

Dependence of the $C_{GTcl}^{2\nu }\left(r\right)$ on the cutoff in the $1^{+}$ excitation energy (top panel) and all $J^{\pi }\mathrm{excitation}$ energies (bottom panel) evaluated for ${}^{76}\mathrm{Ge}$ decay.

Figure 8

The functions $C_{GTcl}^{2\nu }\left(r\right)$ separated into the spin $S=0$ and $S=1$ components, shown for several $\beta \beta$ decay candidate nuclei. The functions were evaluated requiring that $M_{Fcl}^{2\nu }=0$ using the isospin conservation. The values of $M_{GTcl}^{2\nu }\ne 0$ were obtained by choosing the renormalization $g_{pp}^{T=0}$ from the usual condition that the half-life of the $2\nu \beta \beta$ decay is correctly reproduced by the QRPA.

Figure 9

$M^{0\nu }$ evaluated with $g_{pp}^{T=0}$ adjusted such that $M_{cl}^{2\nu }=0$, i.e., partial restoration of the SU(4) symmetry (red circles), or so that the $2\nu \beta \beta$ decay experimental half-lives are correctly reproduced (black squares). For several candidate nuclei $({}^{94}\mathrm{Zr},{}^{110}\mathrm{Pd},{}^{124}\mathrm{Sn}$, and ${}^{134}\mathrm{Xe})$ instead the expected $2\nu \beta \beta$ half-lives were used for the adjustment of the $g_{pp}^{T=0}$ (empty blue squares).