Another look at the impact of an eV-mass sterile neutrino on the effective neutrino mass of neutrinoless double-beta decays

The possible existence of an eV-mass sterile neutrino, slightly mixing with ordinary active neutrinos, is not yet excluded by neutrino oscillation experiments. Assuming neutrinos to be Majorana particles, we explore the impact of such a sterile neutrino on the effective neutrino mass of neutrinoless double-beta decays $\langle m \rangle^\prime_{ee} \equiv m^{}_1 |V^{}_{e1}|^2 e^{{\rm i}\rho} + m^{}_2 |V^{}_{e2}|^2 + m^{}_3 |V^{}_{e3}|^2 e^{{\rm i}\sigma} + m^{}_4 |V^{}_{e4}|^2 e^{{\rm i}\omega}$, where $m^{}_i$ and $V^{}_{ei}$ (for $i = 1, 2, 3, 4$) denote respectively the absolute masses and the first-row elements of the 4$\times$4 neutrino flavor mixing matrix $V$, for which a full parametrization involves three Majorana-type CP-violating phases $\{\rho, \sigma, \omega\}$. A zero effective neutrino mass $|\langle m \rangle^\prime_{ee}| = 0$ is possible no matter whether three active neutrinos take the normal or inverted mass ordering, and its implications for the parameter space are examined in great detail. In particular, given the best-fit values of $m^{}_4 \approx 1.3~{\rm eV}$ and $|V^{}_{e4}|^2 \approx 0.019$ from the latest global analysis of neutrino oscillation data, a three-dimensional view of $|\langle m \rangle^\prime_{ee}|$ in the $(m^{}_1, \rho)$-plane is presented and further compared with that of the counterpart $|\langle m \rangle^{}_{ee}|$ in the absence of any sterile neutrino.