# Neutrino mixing matrix: Can it be derived?

## Main Question or Discussion Point

Hi,

When we consider the two neutrino mixing case, we have the matrix that converts between them as given below

$$\begin{pmatrix} |v_{1}> \\ |v_{2}> \end{pmatrix} = \begin{pmatrix} cos\theta & sin\theta \\ -sin\theta & cos\theta\end{pmatrix}\begin{pmatrix} |v_{e}> \\ |v_{\mu}> \end{pmatrix}$$

Can anyone tell me how to derive this? As far as I know, the only condition that needs to be fulfilled is that the matrix must be unitary (and, because neutrino oscillations do occur, it cannot be the identity matrix). It could even be complex. So why this?

Thank you very much.

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I have seen this kind of matrix on rotation transformation for coordinates. If you have the cartesian coordinate (x, y) then by multiplying them with the above matrix results in a rotation by -$\theta$ radians.

Do you think there's a connection?

Yes, it is the same matrix but I don't know how to connect them. Quite strangely, most sources don't really derive this. So I guess its either easy to derive or an assumption.

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I think the point is that to multiply a mass eigenstate with a phase factor should not change the physical result of neutrino mixing, so we can choose to make the matrix real, whose general form will be like this if it is unitary. This is possible for 2-flavour case since we have 4 phase factors to eliminate in the matrix and can change the phases of 4 eigenstates.

K^2
I don't think there is a theory that predicts the mixing matrix. The only prediction is that the mixing-matrix is non-diagonal only of neutrinos are massive.

So, K^2, you're saying that this is just a hypothesis? Something like let's try this matrix and it works since it agrees with the experiments?

inempty, could you explain why adding or removing phases is okay? Since neutrino oscillation calculations mainly work with the phase, I'm not sure if we can modify them.

So, K^2, you're saying that this is just a hypothesis? Something like let's try this matrix and it works since it agrees with the experiments?

inempty, could you explain why adding or removing phases is okay? Since neutrino oscillation calculations mainly work with the phase, I'm not sure if we can modify them.

I'm not familiar with this topic but I think the key point of neutrino mixing is to write flavour eigenstates in terms of different mass eigenstates and vice versa. But a mass (flavour) eigenstate multiplied a phase factor is still a mass (flavour) eigenstate so It looks that these phase factors won't have impact on the physical result at all.

I've yet to come across a good theory that predicts the neutrino mixing matrix without referring to some other unknowns.

However, the neutrino mixing matrix is related to the neutrino masses and the charged-lepton masses, and the quark masses and mixing angles are a parallel problem.

In the Standard Model and its extensions, the masses of the quarks and leptons are given as mass matrices that connect different generations.

The eigenvalues of these matrices give the masses, but these matrices' eigenvectors need not be aligned, and this misalignment produces cross-generation decay and neutrino mixing.

The next question is what determines these mass matrices. They are the result of interactions with the Higgs particle or some similar mechanism, but that does not really tell us much more about them.

Thus, the Standard Model pushes the problem back a few steps, but does not resolve it:

Elementary-fermion / Higgs Yukawa couplings
->
Elementary-fermion mass matrices
->
Elementary-fermion masses and mixing matrices

Ipetrich, that is rather interesting to know that its not really possible to actually derive it. Even more annoying since the matrix is so simple; its just a rotation matrix after all.

I think that two issues are getting confused here - the form of the mixing matrix and the values of its components. The form of the matrix - in the two generation case, a simple 2x2 rotation matrix - is actually required by the structure of quantum mechanics. Generically, the form of a mixing matrix will be a special unitary matrix, as anything else would change the norm of the vector whose components are being mixed, in conflict with the interpretation of quantum states. Beyond that, overall phases can be ignored, as they will cancel when looking at any observable, and other phases can be absorbed into the definition of the states for a similar reason.

The values of the (independent) components of the mixing matrix on the other hand can only be predicted by a theory that explains the origin of neutrino mass. And, since we do not, yet, have any definitive answers about the origin of neutrino mass, there is nothing, yet, that can make such a prediction from first principles.

Deriving the neutrino mixing matrix? Certainly not in the Standard Model. However, it might be possible to derive that matrix from some GUT. Short of that, I've seen some hypotheses about some of the mass-matrix entries being zero and other such patterns -- "textures".

agree with parlyn. The mixing angle couldn't be determined by theoretical considerations but the form of orthogonal matrix can.

Thank you Parlyne. I think I should have been more explicit; I only wanted to know how the form of the matrix came about. The value of $\theta$ is not what I wanted to know.

Can anyone explain why the structure of QM requires the matrix to be a rotation matrix? And how the general matrix (with the phases included) looks like? I am not too familiar with special unitary groups. Thank you very much for all the help.

Cabibbo–Kobayashi–Maskawa matrix at Wikipedia has a section on "Counting" that explains why.

Let's do the calculation generally, for N generations. The mixing matrix will in general have 2N2 real parameters, since it is N*N and complex. Unitarity forces this down to N2 parameters, and arbitrariness of the phases of the weak and mass eigenstates removes a further (2N-1) parameters. The result:

(N-1)2

This further splits up into (1/2)N(N-1) CP-preserving mixing angles and (1/2)(N-1)(N-2) CP-violating phases.

For 2 generations, there is one CP-preserving angle and no CP-violating phases.
For 3 generations, there are three CP-preserving angles and one CP-violating phase.

Wow, that is exactly what I was looking for. Thank you Ipetrich.

The last piece which I don't understand is why the phases remove (2N-1) parameters. Can you tell me how exactly we get this number? http://math.ucr.edu/home/baez/neutrinos.html explains it somewhat but I am not clear on how multiplying a phase onto the mass and flavour eigenstates "removes" parameters from the matrix. How do we even know what the matrix is like except for the fact that it has four independent parameters?

Thank you very much indeed :)

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Using a general 4 parameter 2x2 unitary matrix
\begin{pmatrix} |v_{1}> \\ |v_{2}> \end{pmatrix} =\begin{pmatrix} e^{\phi 1} e^{\phi 2} & 0 \\ 0 & e^{\phi 1}\end{pmatrix} \begin{pmatrix} cos\theta & sin\theta \\ -sin\theta & cos\theta\end{pmatrix}\begin{pmatrix} e^{\phi 3} & 0 \\ 0 & 1\end{pmatrix} \begin{pmatrix} |v_{e}> \\ |v_{\mu}> \end{pmatrix}
Redefine \begin{pmatrix} |v_{1}> \\ |v_{2}> \end{pmatrix} into
\begin{pmatrix} e^{-\phi 1-\phi 2}|v_{1}> \\ e^{-\phi 2}|v_{2}> \end{pmatrix}
and \begin{pmatrix} |v_{e}> \\ |v_{\mu}> \end{pmatrix} into
\begin{pmatrix} e^{\phi 3}|v_{e}> \\ |v_{\mu}> \end{pmatrix}.

In the redefined basis, we have removed the phases.

Hi,

Just want to revive this thread to ask one more thing. Following the method suggested above by rkrsnan, I realize that there are four possibilites. Rotation in the clockwise and anticlockwise directions and also reflections about the line forming an angle $\theta$ and $-\theta$.

So are all these equally valid mixing matrices? Do we just choose rotation in the counterclockwise direction by $\theta$ as a convention?

Thank you!

EDIT: Let me just write out the matrices

\begin{pmatrix}
\cos\theta & \sin\theta \\
-\sin\theta & \cos\theta
\end{pmatrix}

\begin{pmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{pmatrix}

\begin{pmatrix}
\cos\theta & -\sin\theta \\
-\sin\theta & -\cos\theta
\end{pmatrix}

\begin{pmatrix}
\cos\theta & \sin\theta \\
\sin\theta & -\cos\theta
\end{pmatrix}