# Neutrino mixing

## Homework Statement

In some theory the electron neutrinos and the muon neutrinos mix like this:
$$\mathcal{L}_{m} &= -\frac{1}{2}m\left(\overline{ \nu^{C}_{\mu\text{R}} }\nu_{\text{eL}} + \overline{ \nu^{C}_{\text{eR}} }\nu_{\mu\text{L}}\right) + \text{h.c.}$$

Show that there exists a conserving lepton number in this theory. What are the values for $$\nu_{\text{e}}$$ and $$\nu_{\mu}$$? Are the mass eigenstates Dirac neutrinos or Majorana neutrinos? Does the theory fit with the observational data?

## Homework Equations

$$\mathcal{L}_{m} = \dots = -\frac{1}{2}\left(m\overline{ \nu^{C}_{\mu\text{R}} }\nu_{\text{eL}} + m\overline{ \nu^{C}_{\text{eR}} }\nu_{\mu\text{L}}\right) - \frac{1}{2}\left(m\overline{ \nu_{\text{eL}} }\nu_{\mu\text{R}}^{C}+m\overline{ \nu_{\mu\text{L}} }\nu_{\text{eR}}^{C}\right).$$ Right?

## The Attempt at a Solution

Well, I was just thinking that it must be Dirac type mixing, if there exists a conserving lepton number. So the neutrinos are Dirac neutrinos. Right?

At least, the oscillation fits with the observational data, though in reality there are three types of neutrinos...

But I do not know how to calculate the lepton number. Maybe using the continuity equation and then constructing the Euler-Lagrange equation, but I do not know the exact procedure.

Can you help?

Thanks!