Neutrino oscillations and Majorana Phase

[Soph_the_Oaf]

Hello

Why are neutrino oscillations insensitive to Majorana phases?

I'm guessing it has something to do with them being factored out the PMNS matrix using a diagonal matrix ... i.e.
U_PMNS = U Diag (a1, a2, 1)

Is there a point in the oscillation calculation where they always cancel due to a complex conjugate? You end up with the following in a two-flavour probability

SUM_ij ( Uai U*bi U*aj Ubj )

The majorana phases fall on the diagonals
But in this formula, if i=1, j=2 and b=1, a=2
then you end up with a both the dirac phases in the term

What am I missing? Have we lost the majorana phases before this part?

Cheers

 

[Orodruin]

You are on the right track. You can write the PMNS matrix, including Majorana phases, as
$$
U = V \Phi,
$$
where $V$ is the PMNS matrix with only the Dirac phase and $\Phi$ is a diagonal matrix containing the Majorana phases $e^{i\phi_j}$ (if we count them as three one of them can be absorbed as an overall phase on the charged lepton side, but it does not matter for this argument). Thus, in your probabilities, you will have terms
$$
U_{\alpha i} U^*_{\beta i} U^*_{\alpha j} U_{\beta j} = V_{\alpha i} e^{i\phi_i} V^*_{\beta i}e^{-i\phi_i} V^*_{\alpha j}e^{-i\phi_j} V_{\beta je^{i\phi_j}}
= V_{\alpha i} V^*_{\beta i} V^*_{\alpha j} V_{\beta j}.
$$
It follows that the Majorana phases cannot influence neutrino oscillations.

Another way of seeing the same is that the Hamiltonian is of the form
$$
H = \frac{1}{2E} U M M^\dagger U^\dagger + H_{\rm MSW},
$$
where $M$ is the diagonal matrix with the masses of the mass eigenstates and $H_{\rm MSW}$ is the MSW interaction term. Since $M$ is diagonal, it commutes with $\Phi$ and you obtain
$$
H = \frac{1}{2E} V\Phi M M^\dagger \Phi^\dagger V^\dagger + H_{\rm MSW} = \frac{1}{2E} VM M^\dagger\underbrace{\Phi \Phi^\dagger}_{= \mathbb 1} V^\dagger + H_{\rm MSW} = \frac{1}{2E} VM M^\dagger V^\dagger + H_{\rm MSW},
$$
which again is independent of the Majorana phases.

 

[Soph_the_Oaf]

Thank you very much, that's a really clear explanation! :)
I nearly got to the answer but did some dodgy matrix multiplication in my head, oops. But its nice to have the second explanation you presented also.
Cheers

 

[Soph_the_Oaf]

p.s. for anyone else reading this, a more physical way to think of this is that Majorana phases arise due to the addition of Majorana mass terms that violate lepton number. Neutrino oscillations do not violate lepton number, therefore they are not sensitive to this.