Analytic formulae for T violation in neutrino oscillations

Recently, a concept known as $\mu$TRISTAN, which involves the acceleration of ${\mu }^{+}$, has been proposed. This initiative has led to considerations of a new design for a neutrino factory. Additionally, leveraging the polarization of ${\mu }^{+}$, measurements of T violation in neutrino oscillations are also being explored. In this paper, we present analytical expressions for T violation in neutrino oscillations within the framework of standard three flavor neutrino oscillations, a scenario involving nonstandard interactions, and a case of unitarity violation. We point out that examining the energy spectrum of T violation may be useful for probing new physics effects.