Chiral perturbative reconstruction of the complex orthogonal matrix $R$ in Casas--Ibarra parameterization

In this letter, we perform a chiral perturbative analysis by singular values $m_{Di}$ of the Dirac mass matrix $m_D$ for the type-I seesaw mechanism. In the basis where $m_D=V{m}_D^{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}{U}^{†}$ is diagonal, the mass matrix of right-handed neutrinos $M_R$ is written by $M_R=m_D^{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}{m}^{-1}{m}_D^{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$. If the mass matrix of light neutrinos $m$ has an inverse matrix and the singular values $m_{Di}$ are hierarchical ($m_{D1}\ll m_{D2}\ll m_{D3}$), the singular values $M_i$ and diagonalization matrix $U$ of $M_R$ are obtained perturbatively. By treating $m_{Di}$ and $V$ as input parameters, $m_D$ is represented in the basis where $M_R$ is diagonal, and we perturbatively derive the orthogonal matrix $R$ in Casas–Ibarra parameterization. As a result, $R$ is independent of $m_{Di}$ in the leading order, and it is reconstructed as an orthonormal basis $R_{i1}\simeq \pm \sqrt{m_i\mathrm{/}{m}_{11}}(U_{\mathrm{M}\mathrm{N}\mathrm{S}}^T{V}^{\ast }{)}_{i1},R_{i2}\simeq \pm {\epsilon }_{ijk}{R}_{j3}{R}_{k1},R_{i3}\simeq \pm (U_{\mathrm{M}\mathrm{N}\mathrm{S}}^{†}V{)}_{i3}\mathrm{/}\sqrt{m_i(m^{-1}{)}_{33}}$. Here, $m_i$ is the masses of light neutrinos and $\pm$ denotes the independent degree of freedom for each column vector.