Neutrino Oscillations -partition of PMNS

In summary, the PMNS matrix is written as a multiplication of A, S_ub, S_ol, and M, representing the atmospheric, subleading, solar, and Majorana phase respectively. The dominant angles in each sector are chosen due to the smallness of theta13 and the hierarchy of the mass squares, resulting in good approximations for two flavor oscillations. The mixing matrix can be parameterized using three Euler angles, but this may not apply to all unitary matrices, especially if the neutrinos are Majorana particles where additional phases cannot be removed. However, this does not affect neutrino oscillations.
  • #1
ChrisVer
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I have seen that people write the PMNS matrix as a multiplication of the form:
[itex]\text{PMNS}= A \cdot S_{ub} \cdot S_{ol} \cdot M[/itex]

[itex]\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} [/itex]

where [itex]A[/itex] stands for the atmospheric, [itex]S_{ub}[/itex] is named subleading, [itex]S_{ol}[/itex] for Solar and [itex] M[/itex] 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 [itex]\theta_{23}[/itex] (muon-tau neutrinos) and not the [itex]\theta_{12}[/itex] (muon-electron neutrinos)? Similarily for the Solar part we have only the [itex]\theta_{12}[/itex] (electron-muon neutrinos) and not the [itex]\theta_{13}[/itex] (electron-tau neutrinos)?
Thnaks.
 
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  • #2
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.
 
  • #3
So it's like breaking the SO(3) into three SO(2)?
 
  • #4
If by that you mean parameterising an SO(3) matrix with three Euler angles, then yes.
 
  • #5
I also have one question...
I'm trying to show:
[itex]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}[/itex]

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 [itex]U_{\alpha i}[/itex] to have [itex]U_{i \alpha}[/itex] and same for the rest?

My problem is that if I say:
[itex] | \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[/itex]
then:
[itex] | \nu_i= \sum_{\beta} U_{\beta i}^* |\nu_\beta>[/itex]
However in the book (Carlo Giunti & Chung W. Kim) they write:
[itex]|\nu_\alpha> = \sum_k U_{\alpha k}^* |\nu_k>[/itex]
and
[itex]|\nu_k> = \sum_\beta U_{\beta k} |\nu_\beta>[/itex] (didn't interchange the indices)
 
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  • #6
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.
 
  • #7
Oops.. a typo with the indices... So far it didn't seem to be different...
1: [itex] |\nu_\alpha> = \sum_i U_{\alpha i} |\nu_i> [/itex] and this changes from my previous post: [itex]|\nu_i> = \sum_\alpha U_{i \alpha}^* |\nu_\alpha>[/itex]

vs

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

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 [itex]U_{ei} \ne U_{ie}[/itex].
That's because I'm dragging the transposed indices with:
[itex]|\nu_\alpha (t) > = \sum_{i,\beta} U_{i \alpha} e^{-i E_i t} U_{i \beta}^* |\nu_\beta(0)>[/itex]
whereas the book uses:
[itex]|\nu_\alpha (t) > = \sum_{k,\beta} U_{\alpha k}^* e^{-i E_k t} U_{\beta k} |\nu_\beta(0)>[/itex]
(and OK the difference in complex conjugate is fine, since it's just a difference on how he first defined the unitary transformation)
 
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  • #8
Orodruin said:
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?
[itex]U_{PMNS} = R_{23} R_{13} R_{12}[/itex]
Is it sufficient to say that this combination allows for [itex]U \cdot U^\dagger =1 [/itex]?
 
  • #9
ChrisVer said:
Is there a reason for why you can do that?
[itex]U_{PMNS} = R_{23} R_{13} R_{12}[/itex]
Is it sufficient to say that this combination allows for [itex]U \cdot U^\dagger =1 [/itex]?

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.
 
  • #10
Orodruin said:
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?
 
  • #11
ChrisVer said:
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).
 

What are neutrino oscillations?

Neutrino oscillations are a phenomenon where neutrinos, which are elementary particles, change from one type to another as they travel through space. Neutrinos have three different types or "flavors": electron, muon, and tau. Neutrino oscillations occur because these flavors are not fixed and can change as the neutrino moves.

What is the PMNS matrix?

The PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix is a mathematical representation of the relationship between the three neutrino flavors. It is similar to the CKM matrix used to describe quark flavor mixing. The PMNS matrix is used to describe the probabilities of neutrinos changing from one flavor to another.

What is the significance of neutrino oscillations?

Neutrino oscillations have important implications for our understanding of particle physics and the Standard Model. They also have implications for astrophysics and cosmology, as neutrinos are abundant in the universe and play a role in the evolution of stars and galaxies. Neutrino oscillations also have practical applications, such as in neutrino detectors and potentially in communication technologies.

How are neutrino oscillations studied and measured?

Neutrino oscillations are studied through experiments that observe the interactions of neutrinos. These experiments use large detectors, such as Super-Kamiokande and IceCube, to detect neutrinos and measure their properties. Neutrino oscillations can also be studied indirectly through measurements of other particles, such as muons, that are produced in neutrino interactions.

What is the current understanding of neutrino oscillations?

The current understanding of neutrino oscillations is based on the PMNS matrix, which has been experimentally confirmed by several different experiments. However, there are still many unanswered questions, such as the exact values of the neutrino masses and the reasons for their oscillations. Further research and experiments are needed to fully understand the complexities of neutrino oscillations.

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