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The two-neutrino positron double-$\beta $ decay modes of $^{156}$Dy isotopeare studied in the Projected Hartree-Fock-Bogoliubov framework for the$0^{+}\to 0^{+}$ transition. Theoretically calculated half-lives ofelectron-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.

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