Assumptions

Parameterization

In general, there are nine degrees of freedom in any unitary three by three matrix. However, in the case of the PMNS matrix, five of those real parameters can be absorbed as phases of the lepton fields and thus the PMNS matrix can be fully described by four free parameters.^[6] The PMNS matrix is most commonly parameterized by three mixing angles (${\displaystyle \theta _{12}}$, ${\displaystyle \theta _{23}}$, and ${\displaystyle \theta _{13}}$) and a single phase angle called ${\displaystyle \delta _{\mathrm {CP} }}$ 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:

${\displaystyle {\begin{aligned}&{\begin{bmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{bmatrix}}{\begin{bmatrix}c_{13}&0&s_{13}e^{-i\delta _{\mathrm {CP} }}\\0&1&0\\-s_{13}e^{i\delta _{\mathrm {CP} }}&0&c_{13}\end{bmatrix}}{\begin{bmatrix}c_{12}&s_{12}&0\\-s_{12}&c_{12}&0\\0&0&1\end{bmatrix}}\\&={\begin{bmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta _{\mathrm {CP} }}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta _{\mathrm {CP} }}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta _{\mathrm {CP} }}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta _{\mathrm {CP} }}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta _{\mathrm {CP} }}&c_{23}c_{13}\end{bmatrix}}.\end{aligned}}}$

where ${\displaystyle s_{ij}}$ and ${\displaystyle c_{ij}}$ are used to denote ${\displaystyle \sin \theta _{ij}}$ and ${\displaystyle \cos \theta _{ij}}$ respectively. 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 ${\displaystyle \nu =\nu ^{c}~}$. An infinite number of possible parameterizations exist; one other common example being the Wolfenstein parameterization.

The mixing angles have been measured by a variety of experiments (see neutrino mixing for a description). The CP-violating phase ${\displaystyle \delta _{\mathrm {CP} }}$ has not been measured directly, but estimates can be obtained by fits using the other measurements.

Experimentally measured parameter values

As of November 2022, the current best-fit values from NuFIT.org, from direct and indirect measurements, using normal ordering, are:^[7]

${\displaystyle {\begin{aligned}\theta _{12}&={33.41^{\circ }}_{-0.72^{\circ }}^{+0.75^{\circ }}\\\theta _{23}&={49.1^{\circ }}_{-1.3^{\circ }}^{+1.0^{\circ }}\\\theta _{13}&={8.54^{\circ }}_{-0.12^{\circ }}^{+0.11^{\circ }}\\\delta _{\textrm {CP}}&={197^{\circ }}_{-25^{\circ }}^{+42^{\circ }}\\\end{aligned}}}$

As of November 2022, the 3 σ ranges (99.7% confidence) for the magnitudes of the elements of the matrix were:^[7]

${\displaystyle |U|={\begin{bmatrix}~|U_{\mathrm {e} 1}|~&|U_{\mathrm {e} 2}|~&|U_{\mathrm {e} 3}|\\~|U_{\mu 1}|~&|U_{\mu 2}|~&|U_{\mu 3}|\\~|U_{\tau 1}|~&|U_{\tau 2}|~&|U_{\tau 3}|~\end{bmatrix}}=\left[{\begin{array}{rrr}~0.803\sim 0.845~~&0.514\sim 0.578~~&0.142\sim 0.155~\\~0.233\sim 0.505~~&0.460\sim 0.693~~&0.630\sim 0.779~\\~0.262\sim 0.525~~&0.473\sim 0.702~~&0.610\sim 0.762~\end{array}}\right]}$

Notes regarding the best fit parameter values

• These best fit values imply that there is much more neutrino mixing than there is mixing between the quark flavors in the CKM matrix (in the CKM matrix, the corresponding mixing angles are ${\displaystyle \theta _{12}=\,}$13.04°±0.05° , ${\displaystyle \theta _{23}=\,}$2.38°±0.06° , ${\displaystyle \theta _{13}=\,}$0.201°±0.011° ).
• These values are inconsistent with tribimaximal neutrino mixing (i.e. ${\displaystyle \theta _{12}\approx 35.3^{\circ }\,,}$ ${\displaystyle \theta _{23}=45^{\circ }\,,}$ ${\displaystyle \theta _{13}=0^{\circ }\,}$) at a statistical significance of more than five standard deviations. Tribimaximal neutrino mixing was a common assumption in theoretical physics papers analyzing neutrino oscillation before more precise measurements were available.
• The value of ${\displaystyle \delta _{\textrm {CP}}={197^{\circ }}_{-25^{\circ }}^{+42^{\circ }}}$ is very difficult to measure, and is the object of ongoing research; however the current constraint ${\displaystyle \,{169^{\circ }}\leq \delta _{\textrm {CP}}\leq {246^{\circ }}\,}$ in the vicinity of 180° shows a clear bias in favor of charge-parity violation.