dagger} A W \right\}_{k K} 
\left\{ W^{\dagger} A (UX) \right\}_{K m} 
\left\{ (UX)^{\dagger} A W \right\}_{m L} 
\biggr\}. 
\label{P-beta-alpha-W4-H3-First} \end{aligned}$$

$$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{5th-2nd} 
\equiv 
2 \mbox{Re} \left[ \left( 
S^{(0)}_{\alpha \beta} \right)^{*} 
S_{\alpha \beta}^{(4)} [3]_\text{Second} 
\right] 
\nonumber \\
&=& 
2 \mbox{Re} 
\biggl\{
- \sum_{n} 
\sum_{L k} 
\frac{ 1 }{ \Delta_{L} - h_{k} } 
\left[
(ix) e^{- i ( \Delta_{L} - h_{n} ) x} 
+ 
\frac{ e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} ) } 
\right]
\nonumber \\
&\times& 
W_{\alpha L} (UX)^*_{\beta k} 
(UX)^*_{\alpha n} (UX)_{\beta n} 
\left\{ W^{\dagger} A W \right\}_{L L} 
\left\{ W^{\dagger} A (UX) \right\}_{L k} 
\nonumber \\
&+& 
\sum_{n} 
\sum_{L k} 
\sum_{K \neq L} 
\frac{ 1 }{ ( \Delta_{L} - \Delta_{K} ) ( \Delta_{L} - h_{k} ) ( \Delta_{K} - h_{k} ) } 
\nonumber \\
&\times& 
\biggl[
\left( \Delta_{K} - h_{k} \right) 
e^{- i ( \Delta_{L} - h_{n} ) x} 
- \left( \Delta_{L} - h_{k} \right) 
e^{- i ( \Delta_{K} - h_{n} ) x} 
- \left( \Delta_{K} - \Delta_{L} \right) 
e^{- i ( h_{k} - h_{n} ) x} 
\biggr]
\nonumber \\ 
&\times&
W_{\alpha L} (UX)^*_{\beta k} 
(UX)^*_{\alpha n} (UX)_{\beta n} 
\left\{ W^{\dagger} A W \right\}_{L K}  
\left\{ W^{\dagger} A (UX) \right\}_{K k} 
\nonumber \\ 
&-& 
\sum_{n} 
\sum_{L k} 
\frac{ 1 }{ (\Delta_{L} - h_{k})^2 } 
\biggl[
(ix) \left( e^{- i ( h_{k} - h_{n} ) x} + e^{- i ( \Delta_{L} - h_{n} ) x}  \right)
+ 2 \frac{ e^{- i ( \Delta_{L} - h_{n} ) x} - e^{- i ( h_{k} - h_{n} ) x}  }{ ( \Delta_{L} - h_{k} ) } 
\biggr]
\nonumber \\ 
&\times& 
W_{\alpha L} (UX)^*_{\beta k} 
(UX)^*_{\alpha n} (UX)_{\beta n} 
\left\{ W^{\dagger} A (UX) \right\}_{L k} 
\left\{ (UX)^{\dagger} A W \right\}_{k L} 
\left\{ W^{\dagger} A (UX) \right\}_{L k} 
\nonumber \\ 
&+& 
\sum_{n} 
\sum_{L k} 
\sum_{m \neq k} 
\biggl[
- \frac{ (ix) e^{- i ( \Delta_{L} - h_{n} ) x} }{ ( \Delta_{L} - h_{k} )( \Delta_{L} - h_{m} ) } 
+ \frac{ 1 }{ ( h_{