Correct $\Delta m^2_{ij}$ Dependence for Neutrino Oscillation Formulae

TL;DR

This paper addresses the relativistic invariance issue in neutrino oscillation formulas by proposing the use of proper time and invariant quantities, leading to a correction in the scale of m^2_{ij}.

Contribution

It introduces a relativistically invariant formulation of neutrino oscillation probabilities using proper time instead of energy-based time evolution.

Findings

Using proper time reduces m^2_{ij} scale by a factor of two.

Highlights the importance of relativistic invariance in neutrino oscillation calculations.

Provides a corrected formula consistent with relativistic principles.

Abstract

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 $e^{-imc^2\tau/\hbar}$ where $\tau$ is the proper time of the kaon state. The factor $mc^2\tau$ is the rest frame value of the four vector product $p_\mu x^\mu$ which is an invariant quantity. If $-i p_\mu x^\mu$ is used in neutrino oscillation formulae instead of $-iEt$, the scale of the $\Delta m^2_{ij}$ is reduced by a factor of two.