Path integral of neutrino oscillations

We propose an idea of the constrained Feynman amplitude for the scattering of the charged lepton and the virtual W-boson, $l_{beta} + W_{rho} rightarrow l_{alpha} + W_{lambda}$, from which the conventional Pontecorvo oscillation formula of relativistic neutrinos is readily obtained using plane waves for all the particles involved. In a path integral picture, the neutrino propagates forward in time between the production and detection vertices, which are constrained respectively on the 3-dimensional spacelike hypersurfaces separated by a macroscopic positive time $tau$. The covariant Feynman amplitude is formally recovered if one sums over all possible values of $tau$ (including negative $tau$).