Chiral perturbative analysis for an almost massless neutrino in the type-I seesaw mechanism

In this paper, we perform chiral perturbative analysis of an approximate lepton number symmetry associated with a sufficiently light neutrino in the type-I seesaw mechanism. For the Dirac mass matrix $m_D=(\mathit{A},\mathit{B},\mathit{C})$, linearly independent components of $\mathit{C}$ from $\mathit{A}$ and $\mathit{B}$ are treated as symmetry-breaking parameters. A deviation in the eigenvector of the massless mode $\delta \mathit{u}$ occurs in the first-order perturbation and the lightest mass $m_{1\mathrm{o}\mathrm{r}3}\propto \det m_D^2\mathrm{/}{M}_3$ emerges in the second-order. By solving the perturbation theory, we obtained specific expressions of $\delta \mathit{u}$ and $m_{1\mathrm{o}\mathrm{r}3}$. As a result, two complex parameters in $m_D$ are bounded to some extent from the eigenvector. These constraints are associated with the chiral symmetry and are not susceptible to renormalization.