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 $\mathbb{O}$ in the CI parametrization can be expressed as ${\mathbb{O}}_{ij}^{}=\sqrt{M_j^{}\mathrm{/}{m}_i^{}}{F}_{ij}^{}$ 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 $\mathbb{O}$ is therefore equivalent to imposing some special conditions on the seesaw parameter space.