# Understanding the Mixing Parameters in the PMNS Matrix

I'm trying to understand what the parameters in the PMNS matrix mean exactly. For the two neutrino case the single parameter is a fairly intuitive rotation between basis vectors, however the 3 angles + CP phase in the full PMNS matrix is harder for me to understand.

I have also read that particular experiments are sensitive to individual parameters and would like to know how that works, for example the T2K measurement of muon neutrino disappearance is used to determine the parameter $$\theta_{23}$$.

• You are completely cool with PDG, eqn (14.33) and (14.34)? T2K is treated later in that article. Are you reading this and got stuck on something? Context, please! Feb 23 at 15:33
• @CosmasZachos I don't understand 14.33 and 14.34 very well. In the matrix composition in 14.33, the third matrix from the left looks to me like the matrix for the 2 neutrino case, and the first matrix I imagine is similar but between a different pair of neutrinos but I don't know if this is a correct interpretation. As for the second and fourth I do not know. Feb 23 at 17:50
• WP wisely skips the two Majorana ηs (the 4th matrix) and leaves you with a 4-parameter form, three angles and a phase δ. In your SM course, you learned that, for N Dirac generations, the PMNS matrix has N(N-1)/2 angles and (N-1)(N-2)/2 phases, parameterized thusly. Reducing to the 3 & 1 for N=3, (as opposed to 1 & 0 for N=2). Feb 23 at 19:23

With Wikipedia, let me skip the 4th matrix, diag$$(e^{i\eta_1}, e^{i\eta_2},1)$$, factor of the full 3×3 PMNS matrix (14.33) of the PDG review, so consider the CKM-like matrix as though the neutrinos were Dirac. It is a 3×3 unitary matrix, diagonalizing the mass matrix of the neutrinos in the generation (weak interaction coupling basis) to the mass eigenstate basis, multiplied by the adjoint such for the charged leptons, $$U_{PMNS}= V^\dagger_{l^-}V_\nu.$$ It is unitary, provable directly, since the factors V are unitary!

But it has too much dross: unphysical phases absorbable in the definition of the charged lepton and neutrino spinors. It's more instructive to perform this absorption for N generations. Note in the diagonal mass basis, the W-coupling bilinear $$\bar l {W}_{\!\!L}\!\!\!\!\!\not ~~~~~U\nu$$ has 2N-1 unphysical and absorbable phases. (It would have been 2N, N on each side, but an overall one corresponding to the N×N identity commuting with U is completely invisible and thus disappears on the other side of U before it has to be absorbed by the spinors.)

So the $$N^2$$ parameters of the unitary U can be pruned down to $$N^2 -(2N-1)=(N-1)^2$$ physical ones. This may be parameterized any way you like, e.g. the Wolfenstein parameterization.

Nevertheless, the most memorable one is the one utilizing the SO(N) rotation subgroup of SU(N), with N parameters, henceforth called angles. The trenchant insight of Kobayasi & Maskawa was that for N>2, not all of these parameters could be these angles, and, $$(N-1)^2-N(N-1)/2= (N-1)(N-2)/2$$ would therefore have to be phases! This would jibe for N=2, but would dictate three angles and one (CP!!) phase for N=3.

So you could break down the PMNS parameterization into the three SO(3) rotations, R(23), R(13), R(23) and a rephasing, $$\delta_{\mathrm{CP}}$$, $$U= \begin{bmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{bmatrix} \begin{bmatrix} c_{13} & 0 & s_{13}e^{-i\delta } \\ 0 & 1 & 0 \\ -s_{13}e^{i\delta } & 0 & c_{13} \end{bmatrix} \begin{bmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{bmatrix} \\ = \begin{bmatrix} c_{12}c_{13} & s_{12} c_{13} & s_{13}e^{-i\delta} \\ -s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i\delta } & c_{12}c_{23} - s_{12}s_{23}s_{13}e^{i\delta } & s_{23}c_{13}\\ s_{12}s_{23} - c_{12}c_{23}s_{13}e^{i\delta } & -c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i\delta } & c_{23}c_{13} \end{bmatrix},$$ where $$s_{ij}$$ and $$c_{ij}$$ are used to denote $$\sin\theta_{ij}$$ and $$\cos\theta_{ij}$$, respectively. The middle matrix is just the R(13) rotation flanked by a rephasing and its inverse, $$\operatorname{diag}(e^{-i\delta/2}, 1,e^{i\delta/2})~ R(13)~\operatorname{diag}(e^{i\delta/2}, 1,e^{-i\delta/2})$$.

In the case of Majorana neutrinos, two extra complex phases are needed, as the phase of Majorana fields cannot be freely redefined due to the condition $$\nu = \nu^c$$. So we restore them on the right, the neutrino side, via a compensatory rephasing mentioned at the beginning.

The measurement you are asking about is detailed in this paper.

• Thanks for the detailed answer. Is there any way I can "interpret" the rephasing? I know I can explain the other parts as the rotations of SO(3) and what that means, but I'm less sure I know what the phase is or what it's doing. Mar 3 at 21:27
• It is a group element of the simple unitary group that is not a rotation, and thus diagonal, exp(i(diag(0,-φ,φ)). Mar 5 at 20:20