$\theta_{23}=\pi/4$ and $\delta=-\pi/2$ In Neutrino Mixing, Which Convention?

TL;DR

This paper examines how the values of neutrino mixing angles and CP phase depend on the choice of parameterization convention, emphasizing the importance of specifying the convention when discussing these parameters.

Contribution

It systematically analyzes nine different parameterizations of the neutrino mixing matrix to clarify how mixing angles and phases vary with conventions.

Findings

Mixing angles and phases change with parameterization conventions.

Physical effects remain invariant despite different parameter values.

Explicit relations between different parameterizations are provided.

Abstract

Considerable information has been obtained about neutrino mixing matrix. Present data show that in the particle data group (PDG) parameterization, the 2-3 mixing angle and the CP violating phase are consistent with $\theta_{23} = \pi/4$ and $\delta_{PDG} = -\pi/2$, respectively. A lot of efforts have been devoted to constructing models in realizing a mixing matrix with these values. However, the particular angles and phase are parameterization convention dependent. The meaning about the specific values for mixing angle and phase needs to be clarified. Using the well known 9 independent ways of parameterizing the mixing matrix, we show in detail how the mixing angles and phase change with conventions even with the 2-3 mixing angle to be $\pi/4$ and the CP violating phase to be $-\pi/2$. The original Kaobayashi-Maskawa and an additional one belong to such a category. The other 6 parameterizations have mixing angles and phase very different values from those in the PDG parameterization although the physical effects are the same. Therefore one should give the specific parameterization convention when making statements about values for mixing angles and phase.