Neutrino amplitude decomposition, $S$ matrix rephasing invariance, and reparametrization symmetry

The $S$ matrix rephasing invariance is one of the fundamental principles of quantum mechanics that originates in its probabilistic interpretation. For a given $S$ matrix which describes neutrino oscillation, one can define the two different rephased amplitudes $S_{\alpha \beta}^{ \text{Reph-1} } \equiv e^{ i (\lambda_{1} / 2E) x} S_{\alpha \beta}$ and $S_{\alpha \beta}^{ \text{Reph-2} } \equiv e^{ i (\lambda_{2} / 2E) x} S_{\alpha \beta}$, which are physically equivalent to each other, where $\lambda_{k} / 2E$ denotes the energy eigenvalue of the $k$-th mass eigenstate. We point out that the transformation of the reparametrization (Rep) symmetry obtained with ``Symmetry Finder'' maps $S_{\alpha \beta}^{ \text{Reph-1} }$ to $S_{\alpha \beta}^{ \text{Reph-2} }$, and vice versa, providing a local and manifest realization of the $S$ matrix rephasing invariance by the Rep symmetry of the 1-2 state exchange type. It is strongly indicative of quantum mechanical nature of the Rep symmetry. The rephasing and Rep symmetry relation, though its all-order treatment remains incomplete, is shown to imply absence of the pure 1-3 exchange symmetry in Denton~{\it et al.}~perturbation theory. It then triggers a study of convergence of perturbation series.