Trimaximal mixing

Trimaximal mixing^[1] (also known as threefold maximal mixing^[2]) refers to the highly symmetric, maximally CP-violating, ${\displaystyle 3\times 3}$ fermion mixing configuration, characterised by a unitary matrix (${\displaystyle U}$) having all its elements equal in modulus (${\displaystyle |U_{ai}|=1/{\sqrt {3}}}$, ${\displaystyle a,i=1,2,3}$) as may be written, e.g.:

${\displaystyle U={\begin{bmatrix}{\frac {1}{\sqrt {3}}}&{\frac {1}{\sqrt {3}}}&{\frac {1}{\sqrt {3}}}\\{\frac {\omega }{\sqrt {3}}}&{\frac {1}{\sqrt {3}}}&{\frac {\bar {\omega }}{\sqrt {3}}}\\{\frac {\bar {\omega }}{\sqrt {3}}}&{\frac {1}{\sqrt {3}}}&{\frac {\omega }{\sqrt {3}}}\end{bmatrix}}\Rightarrow (|U_{i\alpha }|^{2})={\begin{bmatrix}{\frac {1}{3}}&{\frac {1}{3}}&{\frac {1}{3}}\\{\frac {1}{3}}&{\frac {1}{3}}&{\frac {1}{3}}\\{\frac {1}{3}}&{\frac {1}{3}}&{\frac {1}{3}}\end{bmatrix}}}$

where ${\displaystyle \omega =\exp(i2\pi /3)}$ and ${\displaystyle {\bar {\omega }}=\exp(-i2\pi /3)}$ are the complex cube roots of unity. In the standard PDG^[3] convention, trimaximal mixing corresponds to: ${\displaystyle \theta _{12}=\theta _{23}=\pi /4}$, ${\displaystyle \theta _{13}=\sin ^{-1}(1/{\sqrt {3}})}$ and ${\displaystyle \delta =\pi /2}$. The Jarlskog ${\displaystyle CP}$-violating parameter ${\displaystyle J}$^[4] takes its extremal value ${\displaystyle |J|=1/(6{\sqrt {3}})}$.

Originally proposed as a candidate lepton mixing matrix,^[5][6] and actively studied^[1][2][7][8] as such (and even as a candidate quark mixing matrix^[9]), trimaximal mixing is now definitively ruled-out as a phenomenologically viable lepton mixing scheme by neutrino oscillation experiments, especially the Chooz reactor experiment,^[10] in favour of the no longer tenable (related) tribimaximal mixing^[11] scheme.