Quadrupolar correlations and deformation effect on two neutrino $\varepsilon \beta ^{+}$ and $\varepsilon \varepsilon $ modes of $^{156}$Dy isotope

Abstract

The two-neutrino positron double-$\beta $ decay modes of $^{156}$Dy isotope are studied in the Projected Hartree-Fock-Bogoliubov framework for the $0^{+}\to 0^{+}$ transition. Theoretically calculated half-lives of electron-positron conversion and double-electron capture modes are presented. The effect of the quadrupolar deformation on nuclear transition matrix element $M_{2\nu}$ is also investigated.