Chiral perturbative solution for the type-I seesaw mechanism in next-to-leading order

In this letter, we perform chiral perturbative diagonalization of the type-I seesaw mechanism by hierarchical singular values $\lambda_{i}$ of the Dirac mass matrix $m_{D}$ up to the next-to-leading order (NLO). Since the mass matrix of right-handed neutrinos $M_{R}$ has parity symmetries under $\lambda_{i} \leftrightarrow - \lambda_{i}$, the singular values $M_{i}$ and mixing angles of diagonalization of $M_{R}$ are written by only even and odd orders of $\lambda_{i}$ respectively. We confirm this fact by specific perturbative expansions of the third- and the fourth-order by $\lambda_{i}$. As a result, as long as the chiral perturbation theory is valid, the NLO contributions are generally suppressed by $O(\lambda_{i}^{2} / \labmda_{j}^{2}) \lesssim 1\%$ compared to the leading-order expressions that have sufficient accuracies.