Parameter symmetries of neutrino oscillations in vacuum, matter, and approximation schemes

Expressions for neutrino oscillations contain a high degree of symmetry, but typical forms the oscillation probabilities mask these symmetries parameters. We elucidate ${2}^{7}$ parameter vacuum parameters and draw connections to choice definitions as well interesting degeneracies. also show that in presence matter an additional set exists. Due complexity exact expressions matter, numerous approximations have been developed; we under certain assumptions approximate at most ${2}^{6}$ include one symmetry related large mixing angle (LMA)-dark degeneracy holds assumption $CPT$ invariance; this adds factor 2 all above cases. Explicit, nontrivial examples are given how physical observables oscillations, such probabilities, $CP$ violation, position solar atmospheric resonance, effective $\mathrm{\ensuremath{\Delta}}{m}^{2}$'s disappearance invariant symmetries. investigate which apply literature more careful consideration improves precision approximations.