Unfinished fabric of the three neutrino paradigm

In the current $3\ensuremath{\nu}$ paradigm, neutrino flavor oscillations probe three mixing angles $({\ensuremath{\theta}}_{12},{\ensuremath{\theta}}_{23},{\ensuremath{\theta}}_{13})$, one $CP$-violating phase $\ensuremath{\delta}$, and two independent differences between the squared masses ${m}_{i}^{2}$, that can be chosen as $\ensuremath{\delta}{m}^{2}={m}_{2}^{2}\ensuremath{-}{m}_{1}^{2}g0$ and $\mathrm{\ensuremath{\Delta}}{m}^{2}={m}_{3}^{2}\ensuremath{-}({m}_{1}^{2}+{m}_{2}^{2})/2$, where $\mathrm{sign}(\mathrm{\ensuremath{\Delta}}{m}^{2})=+(\ensuremath{-})$ for normal (inverted) mass ordering. Absolute $\ensuremath{\nu}$ masses can be probed by the effective mass ${m}_{\ensuremath{\beta}}$ in beta decay, by the total mass $\mathrm{\ensuremath{\Sigma}}$ in cosmology and---if neutrinos are Majorana---by another effective mass ${m}_{\ensuremath{\beta}\ensuremath{\beta}}$ in neutrinoless double beta decay. Within an updated global analysis of oscillation and nonoscillation data, we constrain these $3\ensuremath{\nu}$ parameters, both separately and in selected pairs, and highlight the concordance or discordance among different constraints. Five oscillation parameters $(\ensuremath{\delta}{m}^{2},|\mathrm{\ensuremath{\Delta}}{m}^{2}|,{\ensuremath{\theta}}_{12},{\ensuremath{\theta}}_{23},{\ensuremath{\theta}}_{13})$ are consistently measured, with an overall accuracy ranging from $\ensuremath{\sim}1%$ for $|\mathrm{\ensuremath{\Delta}}{m}^{2}|$ to $\ensuremath{\sim}6%$ for ${\mathrm{sin}}^{2}{\ensuremath{\theta}}_{23}$ (due to its persisting octant ambiguity). We find overall hints for normal ordering (at $\ensuremath{\sim}2.5\ensuremath{\sigma}$), as well as for ${\ensuremath{\theta}}_{23}l\ensuremath{\pi}/4$ and for $\mathrm{sin}\ensuremath{\delta}l0$ (both at 90% C.L.), and discuss some tensions among different datasets. Concerning nonoscillation data, we include the recent KATRIN constraints on ${m}_{\ensuremath{\beta}}$, and we combine the latest $^{76}\mathrm{Ge}$, $^{130}\mathrm{Te}$ and $^{136}\mathrm{Xe}$ bounds on ${m}_{\ensuremath{\beta}\ensuremath{\beta}}$, accounting for nuclear matrix element covariances. We also discuss some variants related to cosmic microwave background (CMB) anisotropy and lensing data, which may affect cosmological constraints on $\mathrm{\ensuremath{\Sigma}}$ and hints on $\mathrm{sign}(\mathrm{\ensuremath{\Delta}}{m}^{2})$. The default option, including all Planck results, irrespective of the so-called lensing anomaly, sets upper bounds on $\mathrm{\ensuremath{\Sigma}}$ at the level of $\ensuremath{\sim}{10}^{\ensuremath{-}1}\text{ }\text{ }\mathrm{eV}$, and further favors normal ordering up to $\ensuremath{\sim}3\ensuremath{\sigma}$. An alternative option, that includes recent ACT results plus other independent results (from WMAP and selected Planck data) globally consistent with standard lensing, is insensitive to the ordering but prefers $\mathrm{\ensuremath{\Sigma}}\ensuremath{\sim}\mathrm{few}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}1}\text{ }\text{ }\mathrm{eV}$, with different implications for ${m}_{\ensuremath{\beta}}$ and ${m}_{\ensuremath{\beta}\ensuremath{\beta}}$ searches. In general, the unfinished fabric of the $3\ensuremath{\nu}$ paradigm appears to be at the junction of diverse searches in particle and nuclear physics, astrophysics and cosmology, whose convergence will be crucial to achieve a convincing completion.