The reason for the update is that there was a recent announcement by the T2K research group of a measurement of anti-muon-neutrinos converting into other anti-neutrino species. I'd like to first show a plot from their recent presentation in which they show the uncertainty in both the 2-3 mixing angle and the 2-3 mass difference.

As can be seen in their figure on Slide 48, the data is entirely consistent with the 2-3 mixing angle being the same for neutrinos as anti-neutrinos. This is a good sign that the mixing angle for anti-neutrinos is the same as for neutrinos; but note that this is also compatible with there being a CP violating phase. In fact, the best fit value for T2K data plus Particle Data Group 2014 data yields a value of the CP violating phase that is close to -90 degrees. What's interesting with these new estimates for the 4 parameters of the PMNS matrix is that (a) the summation of the angles is close to zero (within error) and (b) summation of each of thetas is close to 90 degrees (within error.)

T2K 2015 summary

Angle | (Radians) | (Degrees) | Sin^{2}(θ) |
Sin^{2}(2θ) |

θ_{12} |
0.584 | 33.5 | 0.304 | 0.846 |

θ_{13} |
0.158 | 9.1 | 0.0248 | 0.097 |

θ_{23} |
0.812 | 46.5 | 0.527 | 0.997 |

δ_{CP} |
-1.55 | -88.8 | ||

SUM | 0.00 | 0.0 |

The PMNS corresponding to these values can be visualized using Wolfram Alpha by clicking on the site below:

The 1-sigma uncertainty around the δ

_{CP}term is approximately -90 +/- 90 degrees. This means that there is a good chance that the value of δ

_{CP}is between -180 and 0 degrees, and this also means that there is a really good chance that the exp(i*δ

_{CP}) is non-real, which means that there can be CP violation due to neutrino mixing.

So, using the data above, I've re-evaluated the eigenvalues and determinant of the PMNS maxtrix. I think that the eigenvalues way of viewing this data is better than listing the 4-parameters because one can visualize how close the eigenvalues are to the unit circle. If the eigenvalues fall on the unit circle, then the PMNS matrix is the neutrino mixing is complete (i.e. the mixing is only into these states.) If eigenvalues all fall outside the unit circle, then there is growth in the total number of neutrinos, and if the eigenvalues all fall within the unit circle, then there is decay in the total number of neutrinos.

As seen below, the eigenvalues fall very close to the unit circle (especially if including uncertainty... which is not shown in the figure below...the size of the markers does not correspond to uncertainty in the value of the eigenvalues.) Using previous data, the eigenvalues fall nearly exactly on the unit circle, whereas using the T2K2015+PDG2014 data, the eigenvalues fall slightly off of the unit circle (though, within 1-sigma uncertainty.) Interestingly, one of the eigenvalues is extremely close to 1+0i. The other two eigenvalues are close to the unit circle, but far away from 1+0i. The other thing to point out is that two eigenvalues far from 1+0i are not mirror images of each other on the unit circle. The fact that they are not mirror images is a sign that there is CP violation in the PMNS matrix. If there were no CP violation, one of these eigenvalues would be the complex conjugate of the other one: a +/- bi. The other thing to point is that the value of the determinant is using the new data is nearlly entirely Real valued and slightly less than 1 (Det=0.9505+0.001i). This is likely a sign that the values of the 4-parameters were chosen by T2K in a way that is not consistent, but there is also still the possibility that the non-unitary value of the determinant is due to the fact that there is another type of neutrino that mixes with the three main species. (This is just speculation because, as can be seen from the old wiki data, the eigenvalues fall very close to the unit circle when chosen consistently.)

Old Wiki Data

Angle | (Radians) |

θ_{12} | 0.587 |

θ_{13} | 0.156 |

θ_{23} | 0.670 |

δ_{CP} | -2.89 |

SUM | -1.48 |