这是一份warwick华威大学PX264的成功案例

$$
M=\left(\begin{array}{cc}
0 & m_{D} \
m_{D}^{T} & m_{M}
\end{array}\right)
$$
where $m_{D}$ is a general $3 \times k$ complex matrix and $m_{M}$ is a $k \times k$ complex symmetric matrix. $M$ is diagonalized according to
$$
M_{\text {diag }}=\left(\begin{array}{cc}
m_{\nu} & 0 \
& m_{N}
\end{array}\right)=U M U^{T}
$$
by the (approximate) matrix
$$
U \simeq\left(\begin{array}{cc}
1 & -m_{D} m_{M}^{-1} \
m_{D} m_{M}^{-1} & 1
\end{array}\right)
$$
The seesaw formula for the effective light neutrino mass matrix becomes
$$
m_{\nu} \simeq-m_{D} m_{M}^{-1} m_{D}^{T}
$$
(Minus signs can be removed by field rephasings if necessary.)

PX264 COURSE NOTES ：

$$
E_{i}^{\prime}=E_{f}^{\prime}
$$
has to hold. Thus,
$$
E_{f}=E_{i} \gamma^{2}\left(1-u \cos \theta_{i}\right)\left(1+u \cos \theta_{f}^{\prime}\right),
$$
which leads to
$$
\frac{\Delta E}{E}=\gamma^{2}\left[1-\frac{1}{\gamma^{2}}+u\left(\cos \theta_{f}^{\prime}-\cos \theta_{i}\right)-u^{2} \cos \theta_{i} \cos \theta_{f}^{\prime}\right]
$$