Vacuum Oscillations

Choice of Unit

For numericall purpose, we choose to scale distance x using \(\omega\).

Suppose we use MeV for energy scales, the distance \(\hat x = \omega x\) is related to actually distance \(x\) in km through

\[x = \frac{\hat x}{\omega_v} = \frac{\hat x}{ 1.90\times 10^{-4} \mathrm{m}^{-1} \frac{\delta m^2}{7.5\times 10^{-5}\mathrm{eV}^2} \frac{1\mathrm{MeV}}{E} } = \frac{\hat x}{0.190} \mathrm{km} \frac{7.5\times 10^{-5}\mathrm{eV}^2}{\delta m^2} \frac{E}{1\mathrm{MeV}}.\]

The numericall system is

\[\frac{d\psi}{dx} = -i H \psi,\]

where \(\psi\) is the wave function

\[\begin{split}\psi = \begin{pmatrix} \psi_0 \\ \psi_1 \end{pmatrix}\end{split}\]

and \(H\) is the vacuum Hamiltonian

\[\begin{split}H = \frac{\omega}{2}\begin{pmatrix} -\cos2\theta &\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{pmatrix}.\end{split}\]

Initial condition is set to be

\[\begin{split}\psi(0) = \begin{pmatrix} 1 \\ 0 \end{pmatrix}.\end{split}\]

The theoretical prediction for survival probability of the first flavor is

\[P = 1 - \sin^2 2\theta \sin^2 ( \omega x/2 ).\]