Correct $Δm^2_{ij}$ Dependence for Neutrino Oscillation Formulae
The time translation operator for neutrino mass states is often taken to be $e^{-iEt/\hbar}$. This is not relativistically invariant. In kaon mixing, physicists use $e^{-imc^2τ/\hbar}$ where $τ$ is the proper time of the kaon state. The factor $mc^2τ$ is the rest frame value of the four vector product $p_μx^μ$ which is an invariant quantity. If $-i p_μx^μ$ is used in neutrino oscillation formulae instead of $-iEt$, the scale of the $Δm^2_{ij}$ is reduced by a factor of two.
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