The Generalised Casas-Ibarra Parametrisation for Majorana Neutrino Masses

TL;DR

The paper introduces a Generalised Casas–Ibarra (GCI) parametrisation that encompasses all Majorana neutrino mass models by expressing the light-neutrino mass matrix as $m_\nu=\sum_{i,j} Y_i^T M_{ij} Y_j$ and recasting it as $m_\nu=\mathcal{Y}^T \mathcal{M} \mathcal{Y}$. Through Autonne–Takagi factorisation, the authors diagonalise $\mathcal{M}$ and derive a universal parametrisation $\mathcal{Y}=V^\dagger D^{-1/2}_{\mathcal{M}}\mathcal{R} D_{\sqrt{m}} U^\dagger$, where $\mathcal{R}$ is a (semi)-orthogonal matrix that captures all free neutrino-sector parameters. They apply the framework to representative models (Seesaw, Scotogenic, Linear Seesaw, Generalised Scotogenic, Linear+Inverse Seesaw, Extended Scotogenic) at both tree and loop levels, and provide explicit expressions in well-known cases such as the Zee model with an antisymmetric Yukawa matrix. The approach clarifies model classification, facilitates numerical scans, and yields new insights and parametrisations (e.g., for Zee) with potential extensions beyond the seesaw limit. The work thus offers a unified, flexible toolkit for exploring Majorana neutrino masses across a broad landscape of high-energy models.

Abstract

We present a simple and broadly applicable extension of the Casas-Ibarra parametrisation 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 parametrising 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 Generalised 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.

Figures (5)

• Figure 1: Neutrino mass contributions at the one-loop level in the Scotogenic Model. The neutral scalars $\phi^R_0$ and $\phi_0^I$ run in the loop.
• Figure 2: Neutrino masses at tree level in the Linear seesaw (left) and the Inverse seesaw (right).
• Figure 3: Neutrino masses at the one-loop level in the Generalised Scotogenic Model.
• Figure 4: Neutrino mass contributions at the one-loop level in the Extended Scotogenic Model. Left: Contributions proportional to $m_{L\,(R)}$; Right: Contributions proportional to $m_D$.
• Figure 5: Neutrino masses at the one-loop level in the Zee Model. The mass eigenstates $h^+_{1,2}$ and the charged leptons run in the loop.