Path integral of neutrino oscillations

Abstract

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 $\tau$).