Revisiting series expansions of neutrino oscillation and decay probabilities in matter

We present analytic expressions for three-flavor neutrino oscillations in presence of invisible neutrino decay and matter effects. Using the well-known Cayley-Hamilton formalism, the leading-order terms are derived for oscillation probabilities in all major channels assuming the neutrino mass eigenstate ${\nu }_3$ to decay. Our work extends and complements previous studies utilizing the Cayley-Hamilton theorem, providing the series expansions for ${\nu }_e\to {\nu }_e$, ${\nu }_e\to {\nu }_{\mu }$, ${\nu }_e\to {\nu }_{\tau }$, ${\nu }_{\mu }\to {\nu }_{\mu }$ and ${\nu }_{\mu }\to {\nu }_{\tau }$. The accuracy of the analytical formulas is investigated by comparing the results with numerically calculated probabilities. We also comment on the implications on unitarity violation.