Spin-multipole nuclear matrix elements in thepnquasiparticle random-phase approximation: Implications forβandββhalf-lives

Abstract

Half-lives for 148 potentially measurable 2nd-, 3rd-, 4th-, 5th-, 6th-, and 7th-forbidden unique beta transitions are predicted. To achieve this, the ratio of the nuclear matrix elements (NMEs), calculated by the proton-neutron quasiparticle random-phase approximation (pnQRPA), ${M}_{\mathrm{pnQRPA}}$, and a two-quasiparticle (two-qp) model, ${M}_{\mathrm{qp}}$, is studied and compared with earlier calculations for the allowed Gamow-Teller (GT) ${1}^{+}$ and first-forbidden spin-dipole (SD) ${2}^{\ensuremath{-}}$ transitions. The present calculations are done using realistic single-particle model spaces and $G$-matrix based microscopic two-body interactions. In terms of the ratio $k={M}_{\mathrm{pnQRPA}}/{M}_{\mathrm{qp}}$ the studied decays fall into two groups: for GROUP 1, which consists of transitions involving non-magic nuclei, the ratio turns out to be $k=0.29\ifmmode\pm\else\textpm\fi{}0.15$. For GROUP 2, consisting of transitions involving semimagic nuclei, the ratio is 0.5--0.8 for half of the decays and less than $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ for the other half. The magnitudes of the NMEs for several nuclei of GROUP 2 depend sensitively on the size of the used single-particle space and the energies of few key single-particle orbitals used in the pnQRPA calculation, while no such dependence is found for the transitions involving nuclei of GROUP 1. Comparing the NME ratios $k$ of GROUP 1 with those of the earlier GT and SD calculations, where also experimental data are available, the expected ``experimental'' half-lives for the decays between the ${0}^{+}$ ground state of the even-even reference nuclei and the ${J}^{\ensuremath{\pi}}={3}^{+},{4}^{\ensuremath{-}},{5}^{+},{6}^{\ensuremath{-}},{7}^{+},{8}^{\ensuremath{-}}$ states of the neighboring odd-odd nuclei are derived for possible experimental verification. The present results could also shed light to the magnitudes of the NMEs corresponding to the high-forbidden unique ${0}^{+}\ensuremath{\rightarrow}{J}^{\ensuremath{\pi}}={3}^{+},{4}^{\ensuremath{-}},{5}^{+},{6}^{\ensuremath{-}},{7}^{+},{8}^{\ensuremath{-}}$ virtual transitions taking part in the neutrinoless double beta decays.