# PMNS Matrix parameterization

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The Wikipedia article is not really clear on that. A completely generic n x n unitary matrix has $n^2$ independent parameters, $n(n-1)/2$ of these are angles (i.e. magnitudes) while $n(n+1)/2$ are phases. However, in the case of fermions not all these phases are physical. In fact, in a theory with n generations of leptons you have 2n fields that can be rephased. This means that you can reabsorb $2n-1$ of these phases in a redefinition of the lepton fields. I'm saying $2n-1$ istead of 2n because one of these rephasing (the one when all the phases are the same) is nothing but the conservation of leptonic number and it doesn't influnce the PMNS matrix.

This means that the actual number of physical phases is $n(n+1)/2 - (2n -1) = (n-1)(n-2)/2$ which in your case n=3 gives just one phase.
So, 1 phase and 3 angles give you the 4 independent paramenters.

Note that this is the same procedure used to count the number of CP violating phases of the CKM matrix (which is essentially the quark analogous of the PMNS matrix).

Also note that SU(3) is not the group of ordinary unitary matrices but the group of special unitary matrices. This means one more constraint (det U = +1) and hence 8 independent components instead of 9.

vanhees71
Orodruin
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Let me just add one thing to Einj's excellent reply: It is worth noting that you need ##n-1## additional phases in your parametrization if neutrinos are Majorana in nature. This is due to not being able to rephase Majorana fermions (however, one of the phases is an overall phase which can also be absorbed by rephasing the charged leptons). With three generations you therefore have two additional Majorana phases.

The Majorana phases are irrelevant for oscillation physics but come into play whenever the Majorana nature of the neutrinos would manifest, such as neutrinoless double beta decay. Thus, you will generally not see them mentioned in papers on neutrino oscillations and people will just keep the 4-parameters to describe the PMNS matrix.

vanhees71
Einj, Could you please elaborate a bit more on your argument of rephasing (2n-1) phases instead of 2n?

ohwilleke
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Hi guys,

The wikipedia page on the PMNS matrix talks about there being a way to parametrize any 3x3 (special) unitary matrix by 4 parameters: http://en.wikipedia.org/wiki/Pontecorvo–Maki–Nakagawa–Sakata_matrix#Parameterization

However, I am quite sure SU(3) has dimension 8, it's generators being the 8 Gell-Mann matrices.

Shouldn't you need 8 parameters then?

A unitary four by four matrix has four degrees of freedom and this is enough to fully parameterize Dirac neutrinos. Imagine a three by three matrix where every column and every row must total to one. Fix a two by two block of numbers in any corner. You can calculated the remaining entries with that information. This isn't the way that the PMNS matrix is usually parameterized but it illustrates the point.

In principle, there are two more CP violating degrees of freedom if you have Majorana neutrinos (which is possible because the PMNS matrix for Majorana neutrions doesn't have to be unitary), but experiments won't be good enough to parse out those three CP violating degrees of freedom even if they exist and are actually distinct (just because you have potentially six degrees of freedom doesn't mean that some of them can't be degenerate).

Orodruin
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A unitary four by four matrix has four degrees of freedom and this is enough to fully parameterize Dirac neutrinos. Imagine a three by three matrix where every column and every row must total to one. Fix a two by two block of numbers in any corner. You can calculated the remaining entries with that information. This isn't the way that the PMNS matrix is usually parameterized but it illustrates the point.

This is not correct. To start with, you are likely intending to say three-by-three matrix in the beginning. It is not true that a general 3-by-3 unitary matrix only has four degrees of freedom. It generally has three mixing angles and six complex phases, i.e., nine parameters (it is generally an element of U(3), not SU(3)). Out of these, five of the phases can be removed by rephasing the neutrino and charged lepton fields (if neutrinos are Dirac particles, otherwise only three phases can be removed) and so are unphysical.

The unitary requirement is not that the rows and columns should sum to one, it is that the rows and columns are orthonormal or, in other words, ##U^\dagger U = 1##.

In principle, there are two more CP violating degrees of freedom if you have Majorana neutrinos (which is possible because the PMNS matrix for Majorana neutrions doesn't have to be unitary), but experiments won't be good enough to parse out those three CP violating degrees of freedom even if they exist and are actually distinct (just because you have potentially six degrees of freedom doesn't mean that some of them can't be degenerate).

The mixing matrix for Majorana neutrinos is also unitary, it just has more physical phases. The additional Majorana phases are irrelevant for neutrino oscillation experiments as they are not observable in oscillations. In order to probe them, you need an experiment where the Majorana nature of the neutrinos is manifest, such as neutrinoless double beta decay. It is not that oscillation experiments are not good enough, it is that oscillations as a phenomenon do not depend on the Majorana phases.

ohwilleke
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This is not correct. To start with, you are likely intending to say three-by-three matrix in the beginning. It is not true that a general 3-by-3 unitary matrix only has four degrees of freedom. It generally has three mixing angles and six complex phases, i.e., nine parameters (it is generally an element of U(3), not SU(3)). Out of these, five of the phases can be removed by rephasing the neutrino and charged lepton fields (if neutrinos are Dirac particles, otherwise only three phases can be removed) and so are unphysical.

The unitary requirement is not that the rows and columns should sum to one, it is that the rows and columns are orthonormal or, in other words, ##U^\dagger U = 1##.

The mixing matrix for Majorana neutrinos is also unitary, it just has more physical phases. The additional Majorana phases are irrelevant for neutrino oscillation experiments as they are not observable in oscillations. In order to probe them, you need an experiment where the Majorana nature of the neutrinos is manifest, such as neutrinoless double beta decay. It is not that oscillation experiments are not good enough, it is that oscillations as a phenomenon do not depend on the Majorana phases.

I stand corrected. You are, of course correct that I meant to say 3 by 3 and not 4 by 4. And, I was implicitly assuming real values entries which is only to the extent that there are not complex phases, which isn't a perfect assumption. In the real case, orthonormal and sum to one are the same.

I have previously been told that the mixing matrix for Majorana neutrinos is not unitary, but never understood how that could make sense, so I appreciate your clarification.

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