Ichikawa's wiki

**以前のリビジョンの文書です** ----

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====== 物質効果ありの場合の三世代振動確率 (厳密な取り扱い) ====== O.L.G.Peres, A.Yu.Smirnov Nucl.Phys. B680(2004) 479 "Atmospheric neutrinos:LMA oscillations, Ue3 induced interference and CP-violation"より $\nu_f=\pmatrix{\nu_e\\\nu_\mu\\\nu_\tau}$として、その時間発展は、 <jsmath> i\frac{d\nu_f}{dt}=\pmatrix{\frac{UM^2M^\dagger}{2E}+\hat{V}}\nu_f </jsmath> で与えられる。ここで、$U$はMNS行列$U_{23}D_{\delta_{CP}}U_{13}U_{12}$で、$D_{\delta_{CP}}=\pmatrix{1&0&0\\0&1&0\\0&0&e^{i\delta_{CP}}}$。$V$が、物質によるポテンシャルの項で$V=\pmatrix{V&0&0\\0&1&0\\0&0&1}$である($V=\sqrt{2}G_FN_e$)。 このHamiltonianを対角化して、固有行列$\tilde\nu=\pmatrix{\tilde\nu_e\\ \tilde\nu_\mu\\ \tilde\nu_\tau}$を求めれば良い。 まず、$\nu_f=U_{23}D_{\delta_{CP}}U_{13}\nu'$で与えられる$\nu'$を考える。すると、$\nu'$に対するHamilitonianは、 <jsmath> \begin{eqnarray} H' & = & \frac{1}{2E}U_{12}M^2U_{12}^\dagger+U_{13}VU_{13}\\ & = & \pmatrix{s_{12}^2\delta m_{21}^2+Vc_{13}^2 & s_{12}c_{12}\delta m_{21}^2/2E} \end{eqnarray} </jsmath>