Generalized Casas-Ibarra parametrization for Majorana neutrino masses

We present a simple and broadly applicable extension of the Casas–Ibarra parametrization that captures the structure of all Majorana neutrino mass models. Building directly on the original formulation, our approach naturally accommodates additional degrees of freedom and provides a unified, minimal framework for parametrizing the Yukawa sector. It significantly simplifies both analytical treatments and numerical scans, and can be universally applied to any Majorana neutrino mass model, regardless of the underlying dynamics. The approach also offers a unified framework for classifying neutrino mass models according to the structure of the neutrino mass matrix, which naturally motivates the proposal of an extended version of the scotogenic model. This classification scheme yields tree-level (loop-level) representative models; the seesaw (scotogenic model), the linear seesaw (the generalized scotogenic model), and the linear plus inverse seesaw (the extended scotogenic model). We provide ready-to-use explicit expressions for several well-known scenarios, including the Zee model where one of the Yukawa matrices is antisymmetric.