NEUTRINO PHYSICS

The Mathematics of Neutrino Oscillations

CONSIDER for simplicity two weak eigenstate neutrinos, say $\nu_e$ and $\nu_{\mu}$ . If these neutrinos have masses, then in general these weak eigenstate fields will have the form

$\begin{displaymath}\nu_e = \cos \theta \nu_1 + \sin \theta \nu_2 \end{displaymath}$

(1)

and

$\begin{displaymath}\nu_{\mu} = - \sin \theta \nu_1 + \cos \theta \nu_2, \end{displaymath}$

(2)

where $\nu_1$ and $\nu_2$ are the states with definite masses, m₁ and m₂. If these masses are very small compared with the neutrinos' energy, then weak interactions will produce either $\nu_e$ or $\nu_{\mu}$ . Because these weak eigenstates have two components with different masses, if the state $\nu_e$ is an eigenstate of momentum with eigenvalue p, it follows that $\nu_e$ is a linear combination of two energy eigenstates, H₁ and H₂ where, in natural units ( $h/2 \pi = c = 1$ ),

$\begin{displaymath}H_1 = \left(p_2 + m_1^2 \right)^{1/2} \sim p \left(1 + \frac{m_1}{2 p}^2 \right) \end{displaymath}$

(3)

and

$\begin{displaymath}H_2 = \left(p_2 + m_2^2 \right)^{1/2} \sim p \left(1 + \frac{m_2}{2 p}^2 \right). \end{displaymath}$

(4)

After travelling a distance L, the Schr‡dinger equation says that the two mass eigenstate components gain distinct phases, $\exp \left(iH_1 L \right)$ and $\exp \left(iH_2 L \right)$ . Thus, the neutrino state is no longer a pure $\nu_e$ state and the probability that the state is a $\nu_{\mu}$ state is given by:

$\begin{displaymath}P(\nu_e \to \nu_{\mu}, L) = \left\vert \exp \left( iH_1 L \ri... ...p \left( iH_2 L \right) \sin \theta \cos \theta \right\vert^2. \end{displaymath}$

(5)

Simplifying this equation,

$\begin{displaymath}P(\nu_e \to \nu_{\mu}, L) = \sin^2 2 \theta \sin^2 \left(1.27 L \frac{\delta m^2}{E} \right), \end{displaymath}$

(6)

where $\delta m^2 = m_2^2 - m_1^2$ in units of

eV

², L is the distance in km and E is the neutrino energy in GeV. Thus after travelling a distance L, the initial electron neutrino has now a non-zero probability of becoming a muon neutrino, and the probability oscillates with distance. Note that the largest oscillations occur when $\sin^2 2 \theta = 1$ , otherwise known as maximal oscillations. This phenomenon has actually been quite well studied in the neutral kaon system where the neutral kaon has been observed to oscillate with the neutral antikaon.

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