Sensitive behavior of $2νββ$-decay amplitude within QRPA and broken SU(4) symmetry in nuclei
Making use of an identity transformation independent of a nuclear model, we represent the {\bb}-amplitude as a sum of two terms. One term accounts for most of the sensitivity of the original {\bb}-amplitude to $g'_{pp}$ for realistic $g'_{pp}\simeq 1$ (with $g'_{pp}$ being the ratio of the triplet and singlet p-p interaction strengths) and is determined by a specific energy-weighted sum rule. The sum rule depends only on the particle-particle residual interaction (being linear function of $g'_{pp}$ in the QRPA) and passes through zero at the point $g'_{pp}=1$ where the Wigner SU(4) symmetry is restored in the p-p sector of the Hamiltonian. The second term in the decomposition of the {\bb}-amplitude is demonstrated within the QRPA to be a much smoother function for the realistic values of $g'_{pp}$ than the original {\bb}-amplitude. This term is mainly determined by the intensity of the spin-orbit interaction of the nuclear mean field. Thus, the analysis of the present work reveals the reasons for the sensitivity of the {\bb}-amplitude to different components of the nuclear Hamiltonian and thereby can help in constraining nuclear model uncertainties in calculations of the amplitude.