  By adding a neutrino mass term to the Standard Model, which is Lorentz and
$SU(2)\times U(1)$ invariant but non-local to evade $CPT$ theorem, it is shown
that non-locality within a distance scale of the Planck length, that may not be
fatal to unitarity in generic effective theory, can generate the
neutrino-antineutrino mass splitting of the order of observed neutrino mass
differences, which is tested in oscillation experiments, and non-negligible
baryon asymmetry depending on the estimate of sphaleron dynamics. The one-loop
order induced electron-positron mass splitting in the Standard Model is shown
to be finite and estimated at $\sim 10^{-20}$ eV, well below the experimental
bound $< 10^{-2}$ eV. The induced $CPT$ violation in the $K$-meson in the
Standard Model is expected to be even smaller and well below the experimental
bound $|m_{K}-m_{\bar{K}}|<0.44\times 10^{-18}$ GeV.
