Trimaximal mixing

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= \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 $$\omega=\exp(i2\pi/3)$$ and $$\bar{\omega}=\exp(-i2\pi/3)$$ 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.