\dagger} A W \right\}_{k K} 
\left\{ W ^{\dagger} A W \right\}_{K K} 
\left\{ W ^{\dagger} A (UX) \right\}_{K l} 
\nonumber \\ 
&+& 
\sum_{m} 
\sum_{k \neq l } \sum_{K \neq L } 
\frac{ 1 }{ ( h_{l}  - h_{k} ) ( \Delta_{L} - \Delta_{K} ) ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) ( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} )   } 
\nonumber \\ 
&\times& 
\biggl[ 
( h_{l}  - h_{k} ) 
\biggl\{ ( \Delta_{K} - h_{k} ) ( \Delta_{K} - h_{l} ) 
e^{- i ( \Delta_{L} - h_{m} ) x} - 
( \Delta_{L} - h_{k} ) ( \Delta_{L} - h_{l} ) 
e^{- i ( \Delta_{K} - h_{m} ) x}
\biggr\} 
\nonumber \\ 
&+& 
( \Delta_{L} - \Delta_{K} ) 
\biggl\{ ( \Delta_{K} - h_{k} ) ( \Delta_{L} - h_{k} ) e^{- i ( h_{l} - h_{m} ) x} - 
( \Delta_{K} - h_{l} ) ( \Delta_{L} - h_{l} ) e^{- i ( h_{k} - h_{m} ) x} \biggr\} 
\biggl]
\nonumber \\ 
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l} 
(UX)^*_{\alpha m} (UX)_{\beta m} 
\left\{ (UX)^{\dagger} A W \right\}_{k K} 
\left\{ W ^{\dagger} A W \right\}_{K L} 
\left\{ W ^{\dagger} A (UX) \right\}_{L l} 
\biggr\}.
\label{P-beta-alpha-W4-H3-offdiag}\end{aligned}$$ $$\begin{aligned}
&& P(\nu_\beta \rightarrow \nu_\alpha)^{(4)}_{4th-s} 
\equiv 
2 \mbox{Re} \left[ \left( 
S^{(0)}_{\alpha \beta} \right)^{*} 
S_{\alpha \beta}^{(4)} [4]_{ \text{ offdiag } } (\text{single})
\right]
\nonumber \\
&=& 
2 \mbox{Re} 
\biggl\{
\sum_{n} 
\sum_{k \neq l } 
\sum_{K}
\biggl[
\frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( h_{l} - h_{k} ) } 
e^{ - i ( h_{k} -  h_{n} ) x} 
- 
\frac{ (ix) }{ ( \Delta_{K} - h_{k} )^2 ( \Delta_{K} - h_{l} ) } 
e^{ - i ( \Delta_{K} -  h_{n} ) x} 
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{l} )^2 ( h_{l} - h_{k} )^2 } 
e^{ - i ( h_{l} -  h_{n} ) x}
- \frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( h_{l} - h_{k} )^2 } 
\left( \Delta_{K} + 2 h_{l} - 3 h_{k} \right) 
e^{ - i ( h_{k} -  h_{n} ) x} 
\nonumber \\
&+&
\frac{ 1 }{ ( \Delta_{K} - h_{k} )^3 ( \Delta_{K} - h_{l} )^2 } 
\left( h_{k} + 2 h_{l}  - 3 \Delta_{K} \right) 
e^{ - i ( \Delta_{K} -  h_{n} ) x} 
\biggr]
\nonumber \\ 
&\times&
(UX)_{\alpha k} (UX)^*_{\beta l}
