t\{ W ^{\dagger} A (UX) \right\}_{L k} 
\nonumber \\
&+& 
\sum_{n} \sum_{k} \sum_{K \neq L} \sum_{m \neq k} 
\biggl[
\frac{ (ix) }{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) ( h_{m} - h_{k} ) }
e^{- i ( h_{k} - h_{n} ) x} 
\nonumber \\
&-& 
\frac{1}{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{L} - h_{k} )^2 ( h_{m} - h_{k} )^2 } 
\biggl\{ 
\Delta_{K} \Delta_{L} + ( h_{m} - 2 h_{k} ) ( \Delta_{K} + \Delta_{L} ) + 3 
h_{k}^2 - 2 h_{m} h_{k}
\biggr\}
e^{- i ( h_{k} - h_{n} ) x} 
\nonumber \\
&+&
\frac{1}{ ( \Delta_{K} - h_{m} ) ( \Delta_{L} - h_{m} ) ( h_{m} - h_{k} )^2 } 
e^{- i ( h_{m} - h_{n} ) x} 
\nonumber \\
&+& 
\frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{m} ) } e^{- i ( \Delta_{K} - h_{n} ) x} 
- \frac{1}{ ( \Delta_{K} - \Delta_{L} ) ( \Delta_{L} - h_{k} )^2 ( \Delta_{L} - h_{m} ) } e^{- i ( \Delta_{L} - h_{n} ) x} 
\biggr]
\nonumber \\ 
&\times&
(UX)_{\alpha k} (UX)^*_{\beta k} 
(UX)^*_{\alpha n} (UX)_{\beta n} 
\nonumber \\ 
&\times&
\left\{ (UX)^{\dagger} A W \right\}_{k K} 
\left\{ W ^{\dagger} A (UX) \right\}_{K m} 
\left\{ (UX)^{\dagger} A W \right\}_{m L} 
\left\{ W ^{\dagger} A (UX) \right\}_{L k} 
\biggr\}. 
\label{P-beta-alpha-W4-H4-diag}\end{aligned}$$ $$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{3rd} 
\equiv 
2 \mbox{Re} \left[ \left( 
S^{(0)}_{\alpha \beta} \right)^{*} 
S_{\alpha \beta}^{(4)} [4]_{ \text{ offdiag } } 
\right]
\nonumber \\
&=& 
2 \mbox{Re} 
\biggl\{ 
\sum_{m} 
\sum_{k \neq l } \sum_{K} 
\biggl[ 
- \frac{ (ix) }{ ( \Delta_{K} - h_{k} ) (\Delta_{K} - h_{l} ) } 
e^{- i ( \Delta_{K} - h_{m} ) x} 
+ \frac{ 1 }{  ( h_{l}  - h_{k} ) ( \Delta_{K} - h_{k} )^2  ( \Delta_{K} - h_{l} )^2 } 
\nonumber \\ 
&\times& 
\biggl\{
( h_{l} - h_{k} ) ( h_{l} + h_{k} - 2  \Delta_{K} ) 
e^{- i ( \Delta_{K} - h_{m} ) x} 
+ ( \Delta_{K} - h_{k} )^2 e^{ - i ( h_{l} - h_{m} ) x} 
- ( \Delta_{K} - h_{l} )^2 e^{ - i ( h_{k} - h_{m} ) x} 
\biggr\}
\biggr]
\nonumber \\ 
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l} 
(UX)^*_{\alpha m} (UX)_{\beta m} 
\left\{ (UX)^{