Symmetry Finder: A method for hunting symmetry in neutrino oscillation

Symmetry in neutrino oscillation serves for a better understanding of the physical properties of the phenomenon. We present a systematic way of finding symmetry in neutrino oscillation, which we call Symmetry Finder (SF). By extending the known framework in vacuum into a matter environment, we derive the SF equation, a powerful machinery for identifying symmetry in the system. After learning lessons on symmetry in the Zaglauer-Schwarzer system with a matter equivalent of the vacuum symmetry, we apply the SF method to the Denton et al. (DMP) perturbation theory to first order. We show that the method is so powerful that we uncover the eight reparametrization symmetries with the $1 \leftrightarrow 2$ state exchange in DMP, denoted as IA, IB, $\cdot \cdot \cdot$, IVB, all new except for IA. The transformations consist of the both fundamental and dynamical variables, indicating their equal importance. It is also shown that all the symmetries discussed in this paper can be understood as the Hamiltonian symmetries, which ensures their all-order validity and applicability to varying density matter.