Covert symmetries in the neutrino mass matrix

The flavour neutrino puzzle is often addressed by considering neutrino mass matrices $m$ with a certain number of vanishing entries ($m_{ij}=0$ for some values of the indices), since a reduction in the number of free parameters increases the predictive power. Symmetries that can enforce textures zero can also enforce a more general type of conditions $f(m_{ij})=0$ with $f$ some function of the matrix elements $m_{ij}$. In this case $m$ can have all entries non-vanishing with no reduction in its predictive power. We classify all generation-dependent $U(1)$ symmetries which, in the presence of two leptonic Higgs doublets, can reduce the number of independent high-energy parameters of type-I seesaw to the minimum number compatible with non-vanishing neutrino mixings and CP violation. These symmetries are broken above the scale where the effective operator is generated and can thus remain covert, in the sense that no explicit evidence of the symmetry can be read off the neutrino mass matrix, and different symmetries can give rise to the same low-energy structure. We find that only two cases are viable: one yields a structure with two zero-textures already considered in the literature, the other has no zero-textures and has never been considered before. It predicts normal ordering, a lightest neutrino mass $\sim 10$ meV, a Dirac phase $\delta \sim \frac{3\pi}{2}$ and definite values for the Majorana phases.