Neutrino oscillation - CP violation obscured by matter effect?

[Doofy]

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} )
G_f = Fermi constant
N_e = electron number density in matter
N_n = 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?

 

[AndyW]

Thank you for your question about the matter effect in neutrino oscillation experiments. I can understand your confusion about how the matter effect can mimic CP violation and affect the measured \Delta P. Let me try to provide a brief explanation and point you towards some resources for further reading.

First of all, it is important to understand that CP violation is a fundamental property of the Standard Model of particle physics. It describes the difference between the behavior of particles and their antiparticles under the combined operation of charge conjugation (C) and parity transformation (P). In other words, CP violation means that the laws of physics are not symmetric under the exchange of particles with their antiparticles.

Now, in long baseline neutrino oscillation experiments, we are looking for evidence of CP violation by comparing the oscillation rates of neutrinos and antineutrinos. As you correctly stated, the difference in these rates is given by \Delta P = P(\nu_\alpha \rightarrow \nu_\beta) - P(\overline{\nu}_\alpha - \overline{\nu}_\beta). This difference can be affected by the presence of matter, which introduces a flavor-dependent potential that modifies the neutrino oscillation probabilities.

The reason why the matter effect can mimic CP violation is because it introduces an asymmetry between the behavior of neutrinos and antineutrinos. This is due to the fact that the matter potential affects the neutrinos and antineutrinos differently. In other words, the matter potential breaks the symmetry between particles and antiparticles, just like CP violation does. This is why the matter effect can lead to a difference in the oscillation rates of neutrinos and antineutrinos, which can be mistaken for CP violation.

To see how this works, we can look at the Hamiltonian equations you mentioned in your post. The Hamiltonian describes the evolution of the neutrino flavor states in matter. As you can see, the Hamiltonian for neutrinos and antineutrinos is slightly different, which is due to the presence of the matter potential. This difference in the Hamiltonian leads to different oscillation probabilities for neutrinos and antineutrinos, which can be measured in experiments.

I understand that this may still be a bit confusing, so I would recommend reading some resources that explain the matter effect in more detail. Here are a few suggestions:

1. The paper you mentioned in your post is a good starting point. It