A Catalog of Lepton Mass Textures from Koide and Alikhanov Relations

Abstract

The empirical Koide relation for charged-lepton masses and the recently proposed Alikhanov relation for neutrino mass-squared differences offer intriguing hints toward understanding the structure of lepton mass matrices. Motivated by these observations, we systematically explore charged-lepton and neutrino mass-matrix textures that satisfy these relations and confront their predictions for lepton mixing with experimental data. For the charged-lepton sector, we investigate representative positive Hermitian square-root matrices classified by the number of common eigenvector directions of their generators: three, one, or none. For the neutrino sector, an exhaustive enumeration of real symmetric integer matrices with entries between $-5$ and $5$ identifies six signed-permutation classes with $m_1=0$ and six with $m_1>0$ that satisfy the Alikhanov relation. Combining these neutrino textures with the charged-lepton constructions, allowing for different relative flavor orientations and phase alignments, yields $400$ viable structural pairs consistent with the measured lepton mixing parameters. All retained solutions satisfy the adopted cosmological bound on the sum of neutrino masses and existing laboratory constraints. In particular, the full-rank neutrino textures predict $m_1\simeq16.77$-$17.22\,\mathrm{meV}$. Representative viable solutions yield effective Majorana masses $m_{ββ}\simeq1.29$-$4.03\,\mathrm{meV}$ and $4.07$-$18.67\,\mathrm{meV}$ for the $m_1=0$ and $m_1>0$ branches, respectively, providing potential targets for future high-sensitivity searches.