  In the usual quantum field theoretical approach, neutrino oscillations are
studied diagonalising either the mass or matter Hamiltonians. In this paper we
analyse the problem from an on-shell amplitude perspective, where Lagrangians
or Hamiltonians are not available. We start by studying in detail how flavor
enters in the amplitudes and how the PMNS matrix emerges. We then analyse the
elastic amplitude of two neutrinos and two charged leptons that induce matter
effects and propose a strategy to obtain the known results of the standard
oscillation theory without Hamiltonians. Finally, we extend the previously
proposed procedure and use the most general elastic 4-point amplitude to study
beyond the Standard Model effects on oscillations.
