Physics:Trimaximal mixing

Short description: Fermion mixing configuration

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

$U=\left[\begin{array}{ccc}\frac{1}{\sqrt{3}}& \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{3}}\\ \frac{\omega }{\sqrt{3}}& \frac{1}{\sqrt{3}}& \frac{\overline{\omega }}{\sqrt{3}}\\ \frac{\overline{\omega }}{\sqrt{3}}& \frac{1}{\sqrt{3}}& \frac{\omega }{\sqrt{3}}\end{array}\right]\Rightarrow (|U_{i\alpha }{|}^2)=\left[\begin{array}{ccc}\frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\end{array}\right]$

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

Originally proposed as a candidate lepton mixing matrix, and actively studied as such (and even as a candidate quark mixing matrix), trimaximal mixing is now definitively ruled-out as a phenomenologically viable lepton mixing scheme by neutrino oscillation experiments, especially the Chooz reactor experiment, in favour of the no longer tenable (related) tribimaximal mixing scheme.