Neutrino Oscillations -partition of PMNS

[ChrisVer]

I have seen that people write the PMNS matrix as a multiplication of the form:
\text{PMNS}= A \cdot S_{ub} \cdot S_{ol} \cdot M

\text{PMNS}= \begin{pmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{pmatrix} \cdot \begin{pmatrix} c_{13} & 0 & s_{13} e^{-i \delta} \\ 0 & 1 & 0\\ -s_{13} e^{i \delta} & 0 & c_{13} \end{pmatrix} \cdot \begin{pmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & e^{ia_1/2} & 0 \\ 0 & 0 & e^{ia_2/2} \end{pmatrix}

where A stands for the atmospheric, S_{ub} is named subleading, S_{ol} for Solar and M stands for the Majorana phase.

My question is for example in the "atmospheric" part where we're having the creation of muons and muon-neutrinos, why do we only have the parameter \theta_{23} (muon-tau neutrinos) and not the \theta_{12} (muon-electron neutrinos)? Similarily for the Solar part we have only the \theta_{12} (electron-muon neutrinos) and not the \theta_{13} (electron-tau neutrinos)?
Thnaks.

 

[Orodruin]

You always have all of them. It just so happens that the sectors essentially decouple (due to the smallness of theta13 and the hierarchy of the mass squares) and you are left with one dominating angle in each type of experiment and two flavour approximations being pretty good approximations.

 

[ChrisVer]

So it's like breaking the SO(3) into three SO(2)?

 

[Orodruin]

If by that you mean parameterising an SO(3) matrix with three Euler angles, then yes.

 

[ChrisVer]

I also have one question...
I'm trying to show:
P_{\alpha \rightarrow \beta}(t) = \sum_{i} |U_{\alpha i} U_{\beta i}^* |^2 + 2 \text{Re} \sum_{j>i} U_{\alpha i} U_{\alpha j}^* U_{\beta i}^* U_{\beta j} e^{-i (E_i - E_j) t}

However I have some problem with the indices. Does it really matter whether you obtain the above expression with changing the order of the indices? eg instead of U_{\alpha i} to have U_{i \alpha} and same for the rest?

My problem is that if I say:
| \nu_\alpha > = \sum_i U_{\alpha i} |\nu_i> \Rightarrow |\nu >_F = U |\nu>_M \Rightarrow U^\dagger |\nu>_F = U^\dagger U |\nu>_M = |\nu>_M
then:
| \nu_i= \sum_{\beta} U_{\beta i}^* |\nu_\beta>
However in the book (Carlo Giunti & Chung W. Kim) they write:
|\nu_\alpha> = \sum_k U_{\alpha k}^* |\nu_k>
and
|\nu_k> = \sum_\beta U_{\beta k} |\nu_\beta> (didn't interchange the indices)

 

[Orodruin]

This is largely a matter of how you define the mixing matrix (between fields or between states). Note that the $\beta$ is still the summed quantity so the transposition is there.

 

[ChrisVer]

Oops.. a typo with the indices... So far it didn't seem to be different...
1: |\nu_\alpha> = \sum_i U_{\alpha i} |\nu_i> and this changes from my previous post: |\nu_i> = \sum_\alpha U_{i \alpha}^* |\nu_\alpha>

vs

2(unchanged): |\nu_\alpha> = \sum_k U^*_{\alpha k} |\nu_k> and |\nu_i>= \sum_\beta U_{\beta i} |\nu_\beta>

The thing is that using either 1 or 2, I end up with the similar form for the probability of the transition, but with interchanged indices ... and it doesn't seem right since eg U_{ei} \ne U_{ie}.
That's because I'm dragging the transposed indices with:
|\nu_\alpha (t) > = \sum_{i,\beta} U_{i \alpha} e^{-i E_i t} U_{i \beta}^* |\nu_\beta(0)>
whereas the book uses:
|\nu_\alpha (t) > = \sum_{k,\beta} U_{\alpha k}^* e^{-i E_k t} U_{\beta k} |\nu_\beta(0)>
(and OK the difference in complex conjugate is fine, since it's just a difference on how he first defined the unitary transformation)

 

[ChrisVer]

If by that you mean parameterising an SO(3) matrix with three Euler angles, then yes.

Is there a reason for why you can do that?
U_{PMNS} = R_{23} R_{13} R_{12}
Is it sufficient to say that this combination allows for U \cdot U^\dagger =1?

 

[Orodruin]

Is there a reason for why you can do that?
U_{PMNS} = R_{23} R_{13} R_{12}
Is it sufficient to say that this combination allows for U \cdot U^\dagger =1?

Well, obviously each rotation is unitary in itself (if you also include the phases) so the product must be unitary too. The big question is whether you can parameterise all unitary matrices like this, which you cannot but it is still fine if you remove unphysical phases.

 

[ChrisVer]

which you cannot but it is still fine if you remove unphysical phases

You mean like what you do with CKM matrix, by absorbing phases in the quark fields?

 

[Orodruin]

You mean like what you do with CKM matrix, by absorbing phases in the quark fields?

Yes. The thing with neutrinos is that they may be Majorana in which case you cannot remove as many phases (you get another two in addition to the Dirac phase). This matters for situations when the Majorana nature is manifest, such as neutrinoless double beta decay. It does not affect neutrino oscillations (it is easy to show that the Majorana phases do not affect the oscillation probabilities).