$0\nu \beta \beta$ nuclear matrix elements, neutrino potentials and $\mathrm{SU}(4)$ symmetry

Abstract

Intimate relation between the Gamow-Teller part of the matrix element $M_{\mathrm{GT}}^{0\nu }$ and the $2\nu \beta \beta$ closure matrix element $M_{\mathrm{cl}}^{2\nu }$ is explained and explored. If the corresponding radial dependence $C_{\mathrm{cl}}^{2\nu }(r)$ 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 $M_{\mathrm{cl}}^{2\nu }$ values sensitively depend on the properties 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 $M_{\mathrm{cl}}^{2\nu }$ values. Vanishing values of $M_{\mathrm{cl}}^{2\nu }$ are signs of a partial restoration of the spin-isospin $\mathrm{SU}(4)$ symmetry. We suggest that demanding that $M_{\mathrm{cl}}^{2\nu }$ = 0 is a sensible way, within the method of the Quasi-particle Random Phase Approximation (QRPA), 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 ($\le$ 20\%) from the usual QRPA values when $g_{pp}^{T=0}$ is related to the known $2\nu \beta \beta$ half-lives.

Comments: 12 pages, 9 figures