Constraining the mass-spectra in the presence of a light sterile neutrino from absolute mass-related observables

The framework of three-flavor neutrino oscillation is a well-established phenomenon, but results from the short-baseline experiments, such as the Liquid Scintillator Neutrino Detector (LSND) and MiniBooster Neutrino Experiment (MiniBooNE), hint at the potential existence of an additional light neutrino state characterized by a mass-squared difference of approximately $1\,\rm eV^2$. The new neutrino state is devoid of all Standard Model (SM) interactions, commonly referred to as a 'sterile' state. In addition, a sterile neutrino with a mass-squared difference of $10^{-2}$ $\rm eV^2$ has been proposed to improve the tension between the results obtained from the Tokai to Kamioka (T2K) and the NuMI Off-axis $\nu_e$ Appearance (NO$\nu$A) experiments. Further, the non-observation of the predicted upturn in the solar neutrino spectra below 8 MeV can be explained by postulating an extra light sterile neutrino state with a mass-squared difference around $10^{-5} \rm eV^2$. The hypothesis of an additional light sterile neutrino state introduces four distinct mass spectra depending on the sign of the mass-squared difference. In this paper, we discuss the implications of the above scenarios on the observables that depend on the absolute mass of the neutrinos, namely the sum of the light neutrino masses $(\Sigma)$ from cosmology, the effective mass of the electron neutrino from beta decay $(m_{\beta})$, and the effective Majorana mass $( m_{\beta\beta})$ from neutrinoless double beta decay. We show that some scenarios can be disfavored by the current constraints of the above variables. The implications for projected sensitivity of Karlsruhe Tritium Neutrino Experiment (KATRIN) and future experiments like Project-8, next Enriched Xenon Observatory (nEXO) have been discussed.