  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 $m_{D} = (\bm A \, , \bm B
\, , \bm C)$, linearly independent components of $\bm C$ from $\bm A$ and $\bm
B$ are treated as symmetry-breaking parameters. A deviation in the eigenvector
of the massless mode $\delta \bm u$ occurs in the first-order perturbation and
the lightest mass $m_{1 \, \rm or \, 3} \propto \det m_{D}^{2} / M_{3}$ emerges
in the second-order. By solving the perturbation theory, we obtained specific
expressions of $\delta \bm u$ and $m_{1 \, \rm or \, 3}$. 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.
