\end{array} \right) , \quad
\bar{\nu} = \left(\begin{array}{c}
\bar{\nu}_e \\ \bar{\nu}_\mu \\ \bar{\nu}_\tau \end{array} \right) ,$$ $$H = U \left(\begin{array}{ccc}
0 & 0 & 0 \\ 
0 & \frac{\Delta m^2_{21}}{2 E_\nu} & 0 \\
0 & 0 & \frac{\Delta m^2_{31}}{2 E_\nu} 
\end{array} \right) U^\dagger ,$$ $$M = \left(\begin{array}{ccc}
0 & \mu_{e \mu} & \mu_{e \tau} \\ 
-\mu_{e \mu} & 0 & \mu_{\mu \tau} \\
-\mu_{e \tau} & -\mu_{\mu \tau} & 0
\end{array} \right) ,$$ $$\begin{aligned}
V = -\bar{V} &=& \left(\begin{array}{ccc}
V_{\nu_e} & 0 & 0 \\ 0 & V_{\nu_\mu} & 0 \\ 0 & 0 & V_{\nu_\tau}
\end{array} \right) \\
&=& \frac{\sqrt{2}}{2} G_F \frac{\rho}{m_u}
\left(\begin{array}{ccc}
3Y_e-1 & 0 & 0 \\ 0 & Y_e-1 & 0 \\ 0 & 0 & Y_e-1 \end{array} \right) ,
\nonumber\end{aligned}$$ $$U = \left(\begin{array}{ccc}
c_{12}c_{13} & s_{12}c_{13} & s_{13} \\
-s_{12}c_{23}-c_{12}s_{23}s_{13} & c_{12}c_{23}-s_{12}s_{23}s_{13} &
s_{23}c_{13} \\
s_{12}s_{23}-c_{12}c_{23}s_{13} & -c_{12}s_{23}-s_{12}c_{23}s_{13} &
c_{23}c_{13} \end{array} \right) ,$$ $\Delta m^2_{ij} = m^2_i - m^2_j$, $m_i$ is the mass of the $i$th mass eigenstate neutrinos, $E_\nu$ is the neutrino energy, $G_F$ is the Fermi constant, $\rho$ is the density, $m_u$ is the atomic mass unit, $\mu_{\alpha \beta}$ is the transition neutrino magnetic moment, $B_\bot$ is the transverse magnetic field, $s_{ij} = 