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 $m_{D} = V m_{D}^{\rm diag} U^{\dagger}$ is diagonal, the mass matrix of right-handed neutrinos $M_{R}$ is written by $M_{R} = m_{D}^{\rm diag} m^{-1} m_{D}^{\rm diag}$. 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 $R_{i1} \simeq \pm \sqrt{m_{i} / m_{11} } (U_{\rm MNS}^{T} V^{*})_{i1} \, , R_{i2} \simeq \pm \epsilon_{ijk} R_{j3} R_{k1} \, , R_{i3} \simeq \pm {(U_{\rm MNS}^{\dagger} V)_{i3} / \sqrt {m_{i} (m^{-1})_{33}} }$. Here, $m_{i}$ is the masses of light neutrinos and $\pm$ denotes the independent degree of freedom for each column vector.