On the orthogonal matrix in the Casas-Ibarra parametrization for the Yukawa interactions of Majorana neutrinos

The Casas-Ibarra (CI) parametrization of the Yukawa coupling matrix of Majorana neutrinos is generalized by considering the exact seesaw relation and including non-unitarity of the $3\times 3$ Pontecorvo-Maki-Nakagawa-Sakata (PMNS) flavor mixing matrix. With the help of a full $6\times 6$ Euler-like block description of the flavor structure for the seesaw mechanism, we find that the orthogonal matrix $O$ in the CI parametrization can be expressed as $O_{ij}=\sqrt{M_j/m_i}{F}_{ij}$ with $m_i$ and $M_j$ being the masses of light and heavy Majorana neutrinos and $F_{ij}$ consisting of the PMNS and active-sterile flavor mixing parameters (for $i,j=1,2,3$). Assuming a specific pattern of $O$ is therefore equivalent to imposing some special conditions on the seesaw parameter space.