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

Intimate relation between the Gamow-Teller part of the matrix element $M^{0\nu}_\mathrm{GT}$ and the $2\nu\beta\beta$ closure matrix element $M^{2\nu}_\mathrm{cl}$ is explained and explored. If the corresponding radial dependence $C^{2\nu}_\mathrm{cl}(r)$ 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 $M^{2\nu}_\mathrm{cl}$ 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 $M^{2\nu}_\mathrm{cl}$ values. Vanishing values of $M^{2\nu}_\mathrm{cl}$ are signs of a partial restoration of the spin-isospin $\mathrm{SU}(4)$ symmetry. We suggest that demanding that $M^{2\nu}_\mathrm{cl}$ = 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 $g^{T=0}_{pp}$. 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 $g^{T=0}_{pp}$ is related to the known $2\nu\beta\beta$ half-lives.