# Pontecorvo–Maki–Nakagawa–Sakata matrix

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In particle physics, the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix), Maki–Nakagawa–Sakata matrix (MNS matrix), lepton mixing matrix, or neutrino mixing matrix, is a unitary matrix[note 1] which contains information on the mismatch of quantum states of leptons when they propagate freely and when they take part in the weak interactions. It is important in the understanding of neutrino oscillations. This matrix was introduced in 1962 by Ziro Maki, Masami Nakagawa and Shoichi Sakata,[1] to explain the neutrino oscillations predicted by Bruno Pontecorvo.[2][3]

## The matrix

For three generations of leptons, the matrix can be written as:

$\begin{bmatrix} {\nu_e} \\ {\nu_\mu} \\ {\nu_\tau} \end{bmatrix} = \begin{bmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \end{bmatrix} \begin{bmatrix} \nu_1 \\ \nu_2 \\ \nu_3 \end{bmatrix}. \$

On the left are the neutrino fields participating in the weak interaction, and on the right is the PMNS matrix along with a vector of the neutrino fields diagonalizing the neutrino mass matrix. The PMNS matrix describes the probability of a neutrino of given flavor α to be found in mass eigenstate i. These probabilities are proportional to |Uαi|2.

Various parametrizations of this matrix exist,[4] however due to the difficulties of detecting neutrinos, it is much more difficult to determine the individual coefficients than in the equivalent matrix for the quarks (the CKM matrix). The PMNS matrix is most commonly parameterized by three mixing angles (θ12, θ23 and θ13) and a single phase called δCP related to charge-parity violations (i.e. differences in the rates of oscillation between two states with opposite starting points which makes the order in time in which events take place necessary to predict their oscillation rates), in which case the matrix can be written as:

\begin{align} & \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_{13}} \\ 0 & 1 & 0 \\ -s_{13}e^{i\delta_{13}} & 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_{13}} \\ -s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i\delta_{13}} & c_{12}c_{23} - s_{12}s_{23}s_{13}e^{i\delta_{13}} & s_{23}c_{13}\\ s_{12}s_{23} - c_{12}c_{23}s_{13}e^{i\delta_{13}} & -c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i\delta_{13}} & c_{23}c_{13} \end{bmatrix}. \end{align}

where sij and cij are used to denote sinθij and cosθij respectively.

The mixing angles have been measured by a variety of experiments (see neutrino mixing for a description). The CP-violating phase δCP has not been measured directly, but estimates can be obtained by fits using the other measurements. As of July 2014, the current best-fit values from NuFit are:[5][6]

\begin{align} \theta_{12} [^\circ]& = 33.36^{+0.81}_{-0.78} \\ \theta_{23} [^\circ] & = 40.0^{+2.1}_{-1.5}~\textrm{or}~50.4^{+1.3}_{-1.3} \\ \theta_{13} [^\circ] & = 8.66^{+0.44}_{-0.46} \\ \delta_{\textrm{CP}} [^\circ] & = 300^{+66}_{-138} \\ \end{align}