On the two-loop radiative origin of the smallest neutrino mass and the associated Majorana CP phase

Given a massless neutrino at a superhigh energy scale $\Lambda$ (e.g., in the minimal seesaw model with only two heavy Majorana neutrinos), we calculate quantum corrections to its initially vanishing mass $m^{}_1$ (or $m^{}_3$) and the associated Majorana CP phase $\rho$ (or $\varrho$) at the Fermi scale $\Lambda^{}_{\rm F}$ by means of the two-loop renormalization-group equations (RGEs) in the standard model and with the help of the latest neutrino oscillation data. The numerical results obtained from our analytical approximations are in good agreement with those achieved by numerically solving the two-loop RGEs. In particular, we confirm that a nonzero value of $m^{}_1$ (or $m^{}_3$) of ${\cal O}(10^{-13})$ eV at $\Lambda^{}_{\rm F}$ can be radiatively generated from $m^{}_1 =0$ (or $m^{}_3 =0$) at $\Lambda \simeq 10^{14}$ GeV in the SM, and find that $\rho$ (or $\varrho$) may accordingly acquire an appreciable physical value. As a nontrivial by-product, the evolution of all the other (initially nonzero) flavor parameters of massive neutrinos is studied both analytically and numerically, by just keeping their leading (i.e., one-loop) RGE-induced effects.