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 SReph-1 (alpha beta) = e(i(lambda 1/2E)x)S(alpha beta) and SReph-2 (alpha beta )= 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(Reph-1 ) to SReph-2 (alpha beta), 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 et al. perturbation theory. It then triggers a study of convergence of perturbation series.