(\lambda_{m} -\lambda_{k} ) }, 
\label{S-alpha-beta-0th-final}\end{aligned}$$ and the oscillation probability by $P(\nu_\beta \rightarrow \nu_\alpha) = \vert S_{\alpha \beta}^{(0)} \vert^2$.

Finally, armed with the solution , we can also calculate all higher order terms in oscillation probability for e.g. those in eq.  since only such combination $X_{ik} X_{jk}^*$ (no sum over $k$ implied) can appear.

 Low-scale vs. high-scale unitarity violation and $W$ corrections {#sec:UV-low-high-E}
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 Low-scale versus high-scale unitarity violation  {#sec:low-vs-high}
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In leptonic unitarity test, a clear understanding of the relationship between low-scale and high-scale unitarity violation may be one of the key issues. While the presence of probability leaking term $\mathcal{C}_{\alpha \beta}$, if detected, clearly testifies for low-scale unitarity violation [@Fong:2016yyh], it would be better to provide a global view of the difference between them, looking for alternative ways to reach the goal. In this section, we would like to give a preliminary discussion on this topics.

One would guess, on intuitive ground, that at zeroth order in $W$ our system describes high-scale unitarity violation. There is no “$W$ corrections” in high-scale unitarity violation because the energy scale is so high that the high-mass sector is truncated. It is in agreement with the formulations in ref. [@Blennow:2016jkn] with which we share the same evolution equation (\[Schroedinger-eq-0th\]) in the vacuum mass eigenstate basis. See also [@Antusch:2006vwa].[^16] Based on this perspective, we just conclude this subsection by stating the generic characteristic differences between high-scale and low-scale unitarity violation:

-   Our system calculated with evolution equations (\[Schroedinger-eq-0th\]) in leading (zeroth) order in $W$ expansion, which is common to both high- and low-scale unitarity violation, would serve as a firs