Radiative Breaking of Two-Zero Neutrino Mass Minors: Revisiting the $\mathrm{U}(1)_{L_\mu-L_\tau}$ Model
The two-zero minor structure predicted by flavor symmetries is usually discussed as a tree-level relation among low-energy neutrino parameters. We point out that this relation can be significantly modified by radiative corrections even when the two-zero minor structure is enforced by an underlying symmetry at tree level. As a concrete example, we analyze the minimal $\mathrm{U}(1)_{L_\mu-L_\tau}$ model and compute the universal one-loop threshold corrections associated with the type-I seesaw sector. These corrections generate flavor-dependent contributions to the Weinberg operator and violate the exact two-zero minor conditions once the $\mathrm{U}(1)_{L_\mu-L_\tau}$ symmetry is spontaneously broken. This effect relaxes the tree-level lower bound on the total neutrino mass and thereby weakens the tension between the model and cosmological neutrino-mass constraints.