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.
Categories: hep-ph hep-th