approximation of ignoring the fourth (and higher-order) corrections, they are identical to $\delta P(\nu_{\mu} \rightarrow \nu_{\alpha})  \equiv P(\nu_{\mu} \rightarrow \nu_{\alpha}) - P(\nu_{\mu} \rightarrow \nu_{\alpha})^{(0)}$. In each panel the three cases are examined. $N=1$ case with maximal $\mathcal{C}_{\mu \alpha}$ (solid line), the universal scaling model with $N=3$ (dotted line), and the order $W^2$ correction only (dashed line). The last case corresponds to the universal scaling model with $N=\infty$. The blue lines are for $E=10$ GeV, and the red for $E=100$ GeV.

We will first focus on the appearance channels $\nu_\mu \to \nu_e$ and $\nu_\mu \to \nu_\tau$. At $E=10$ GeV (100 GeV) $\delta P$ depends very much on the above three cases, $N=1$, $N=3$, and $N=\infty$ for baseline $L$ of several 100 km ($L \gsim 1000$ km). The maximum value of $| \delta P |$ is always given by the case of maximal (minimal) $\mathcal{C}_{\mu \alpha}$ for positive (negative) $\delta P(\nu_{\mu} \rightarrow \nu_{\alpha})$ shown by the solid (dashed) lines. These maximal values of $| \delta P(\nu_{\mu} \rightarrow \nu_{\alpha}) |$ are, roughly speaking, $\simeq 10^{-3}$ for $\nu_{\mu} \rightarrow \nu_{\tau}$, and $\simeq \text{a few} \times 10^{-4}$ for $\nu_{\mu} \rightarrow \nu_{e}$. The effect might be visible for the former, though it might be challenging for the latter channel.[^21] At longer distance, however, we see enhancement. At $E=10$ GeV, we observe a factor of several enhancement both in $| \delta P (\nu_{\mu} \rightarrow \nu_{\alpha})|$ for $\nu_{\mu} \rightarrow \nu_{e}$ and $\nu_{\mu} \rightarrow \nu_{\tau}$ in region $L \gsim 3000$ km, which may provide a clearer signature. The similar tendency exists at $E=100$ GeV, but in a less pronounced way. On the other hand, $\delta P(\nu_{\mu} \rightarrow \nu_{\alpha})$ flips sign at around $1000-3000$ km for $\nu_{\mu} \rightarrow \nu_{e}$, and $3000-6000$ km for $\nu_{\mu} \rightarrow \nu_{\tau}$ channels. It produces, assuming detector’s sensitivity, a peculiar z