First let us discuss the case where time evolution is unitary. From Equation (10), we have Furthermore, from Equation (16), the factor $\mathrm{Im}\left[{\tilde{X}}_2^{e\mu }{\tilde{X}}_3^{e\mu *}\right]$ in Equation (17) can be rewritten as Equations (17) and (18) are applicable for a generic case, as long as unitarity relation (11) holds.

The discussions in Section 3.1 are based on the assumption that time evolution is unitary. In Ref. [42], the possibility to have a non-unitary leptonic mixing matrix was pointed out. In that case, the relation between the mass eigenstate ${\nu }_j$ and the flavor eigenstate ${\nu }_{\alpha }$ is given by a nonunitary matrix N:

with In the so-called minimal unitarity violation, which was discussed in Ref. [42], the constraint on the deviation matrix $\eta$ turned out to be strong. Here we take phenomenologically the form of the nonunitary matrix N and assume that the elements of the deviation matrix $\eta$ is of order $\Delta E_{21}/\Delta E_{31}$ or smaller, as in Section 3.1.2, namely, It was argued in Ref. [43] that time evolution in the case of nonunitary mixing matrix can be discussed in terms of the mass eigenstate and its time evolution is described by where $\mathcal{E}$ and $\mathcal{A}$ are defined by Equations (6) and (7), stands for the absolute value of the contribution to the matter effect from the neutral current interaction, and the term $A_n\mathbf{1}$ was added to simplify the calculations without changing the absolute value of the probability amplitude. The $3\times 3$ matrix on the right hand side of Equation (30) can be diagonalized with a unitary matrix W: where is the energy eigenvalue matrix in matter with unitarity violation. The mass eigenstate at distance L can be solved as In cases involving unitarity violation, due to the modified form of the charged current interaction [42], after computing the probability amplitude from Equation (32), we must multiply the probability amplitude by an additional factor of ${\left(N{N}^{†}\right)}_{\beta \beta }^{-1/2}$ for the production process and ${\left(N{N}^{†}\right)}_{\alpha \alpha }^{-1/2}$ for detection. Defining the modified amplitude the modified probability and the quantity we have the following expression for the appearance oscillation probability: T violation $P({\nu }_{\mu }\to {\nu }_e)-P({\nu }_e\to {\nu }_{\mu })$ is a small quantity, and the difference between the probability $P({\nu }_{\mu }\to {\nu }_e)$ and the modified one $\widehat{P}({\nu }_{\mu }\to {\nu }_e)$ comes from the factor ${\left(N{N}^{†}\right)}_{\alpha \alpha }{\left(N{N}^{†}\right)}_{\beta \beta }={\left[{(\mathbf{1}+\eta )}^2\right]}_{\alpha \alpha }{\left[{(\mathbf{1}+\eta )}^2\right]}_{\beta \beta }\simeq 1+2{\eta }_{\alpha \alpha }+2{\eta }_{\beta \beta }$, which has a small deviation from 1. Therefore, T violation of the probability $P({\nu }_{\mu }\to {\nu }_e)-P({\nu }_e\to {\nu }_{\mu })$ can be approximated by that of the modified probability $\widehat{P}({\nu }_{\mu }\to {\nu }_e)-\widehat{P}({\nu }_e\to {\nu }_{\mu })$. Hence T violation is given by   We observe that the energy dependence of T violation in this case is different from that with unitarity, since we have extra contributions which are proportional to $sin\left(\Delta {\tilde{E}}_{31}L\right)$ or $sin\left(\Delta {\tilde{E}}_{21}L\right)$. As in the case with unitarity, ${\tilde{\mathcal{X}}}_j^{\alpha \beta }$ can be expressed in terms of the quantity $X_j^{\alpha \beta }\equiv U_{\alpha j}{U}_{\beta j}^{*}$ in vacuum, ${\tilde{E}}_j$, and ${\eta }_{\alpha \beta }$. First of all, we note the following relations: Then we rewrite Equations (35) as where $V_{jm}\equiv {\left({\tilde{E}}_m\right)}^{j-1}$ is the element of the Vandermonde matrix V as in the case with unitarity (See Equation (15)). The simultaneous Equation (36) can be solved by inverting V and we obtain The factor $\mathrm{Im}\left[{\tilde{\mathcal{X}}}_2^{e\mu }{\tilde{\mathcal{X}}}_3^{e\mu *}\right]$ can be expressed in terms of ${\tilde{E}}_j$ and ${\mathcal{Y}}_j^{e\mu }$:   The quantities ${\mathcal{Y}}_j^{e\mu }(j=1,2,3)$ are calculated as follows: In the current scenario involving unitarity violation, we observe a nonvanishing contribution from ${\mathcal{Y}}_1^{\alpha \beta }$, necessitating knowledge of the explicit form of the energy eigenvalue ${\tilde{E}}_1$. Given that ${\mathcal{Y}}_1^{e\mu }$ is of order $O\left({\eta }_{\alpha \beta }\right)$, to evaluate Equation (38) accurately to first order in both $\Delta E_{21}/\Delta E_{31}$ and ${\eta }_{\alpha \beta }$, we must calculate ${\tilde{E}}_1$ solely to zeroth order in these parameters, i.e., assuming $\Delta E_{21}\to 0$ and ${\eta }_{\alpha \beta }\to 0$. Under these conditions, the characteristic equation of the $3\times 3$ matrix (31) is defined by   where ${\lambda }_{\pm }$ are the roots of the quadratic equation and are given by From this, we obtain the energy eigenvalues ${\tilde{E}}_j(j=1,2,3)$ to the leading order in $\Delta E_{21}/\Delta E_{31}$ and ${\eta }_{\alpha \beta }$: The roots ${\lambda }_{-}={\tilde{E}}_1$ and ${\lambda }_{+}={\tilde{E}}_3$ satisfy of the quadratic equation Hence the first two term on the right hand side of Equation (38) can be rewritten as whereas the third term on the right hand side of Equation (38) can be written as Thus we obtain the expression for the factor $\mathrm{Im}\left[{\tilde{\mathcal{X}}}_2^{e\mu }{\tilde{\mathcal{X}}}_3^{e\mu *}\right]$:   To complete the calculation of Equation (34), we need to estimate the two quantities: where terms of order $O\left({(\Delta E_{21}/\Delta E_{31})}^2\right)$, $O\left({\left({ϵ}_{\alpha \beta }\right)}^2\right)$ and $O({ϵ}_{\alpha \beta }\Delta E_{21}/\Delta E_{31})$ have been neglected in Equations (39)–(41). Putting Equations (39)–(41) together, the final expression for T violation is given by Due to the additional contribution proportional to $sin(\Delta {\tilde{E}}_{31}L)$, the energy dependence of Equation (42) in the scenario with unitarity violation differs from that in the scenarios with unitarity, such as the standard case (23) and the nonstandard interaction case (42). Therefore, if the contribution from unitarity violation is significant enough and the experimental sensitivity is sufficiently high, it may be possible to distinguish the unitarity violation scenario from both the standard and nonstandard interaction scenarios by examining the energy spectrum in T violation.