## Neutrino oscillation - CP violation obscured by matter effect?

In long baseline neutrino oscillation experiments, it is possible to investigate the extent of any CP violation by looking at the difference between the rate of neutrino oscillating vs. anti-neutrinos oscillating, ie. we take $\Delta P = P(\nu_\alpha \rightarrow \nu_\beta) - P(\overline{\nu}_\alpha - \overline{\nu}_\beta)$.

The neutrinos propagate through the Earth's crust, which introduces a flavour-dependent matter potential which affects the measured parameters (ie. the mixing angles and mass splittings in matter are slightly different to those in vacuum). This much I understand, as I have been able to find an understandable derivation of equations which relate the in-vacuum parameters to the in-matter parameters.

However, what I do not understand is how the matter effect is able to mimic CP violation and affect the measured $\Delta P$. Having googled around, the best hint I have found is this difference in Hamiltonian between neutrino and antineutrino:

$H_\nu = \frac{1}{2p}(UM^{2}U^{\dagger} + diag(a_{cc}, 0, a_{nc}))$
$H_{\overline{\nu}} = \frac{1}{2p}(U^{*}M^{2}U^{T} - diag(a_{cc}, 0, a_{nc}))$

where:
U = PMNS matrix
$a_{cc} = 2\sqrt{2}G_fN_ep$
$a_{nc} = \sqrt{2}G_fN_np$
M^{2} = diag(m_{1}^{2}, m_{2}^{2}, m_{3}^{2}, m_{4}^{2} )
Gf = Fermi constant
Ne = electron number density in matter
Nn = neutron number density in matter
p = momentum

The paper I got this from (http://arxiv.org/pdf/hep-ph/9712537v1.pdf) just states this rather than giving an explanation as far as I can tell. So, my question is, can anyone please show me / point me towards a derivation of this?

Also, this is done for 4 neutrino mass states - does a simpler treatment for just 3 neutrinos exist? Or is four the minimum required for some reason?
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