Trimaximal mixing (also known as threefold maximal mixing) refers to the highly symmetric, maximally CP-violating, 3 × 3 {\displaystyle 3\times 3} fermion mixing configuration, characterised by a unitary matrix ( U {\displaystyle U} ) having all its elements equal in modulus ( | U a i | = 1 / 3 {\displaystyle |U_{ai}|=1/{\sqrt {3}}} , a , i = 1 , 2 , 3 {\displaystyle a,i=1,2,3} ) as may be written, e.g.:
U = [ 1 3 1 3 1 3 ω 3 1 3 ω ¯ 3 ω ¯ 3 1 3 ω 3 ] ⇒ ( | U i α | 2 ) = [ 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 ] {\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 ω = exp ( i 2 π / 3 ) {\displaystyle \omega =\exp(i2\pi /3)} and ω ¯ = exp ( − i 2 π / 3 ) {\displaystyle {\bar {\omega }}=\exp(-i2\pi /3)} are the complex cube roots of unity. In the standard PDG convention, trimaximal mixing corresponds to: θ 12 = θ 23 = π / 4 {\displaystyle \theta _{12}=\theta _{23}=\pi /4} , θ 13 = sin − 1 ( 1 / 3 ) {\displaystyle \theta _{13}=\sin ^{-1}(1/{\sqrt {3}})} and δ = π / 2 {\displaystyle \delta =\pi /2} . The Jarlskog C P {\displaystyle CP} -violating parameter J {\displaystyle J} takes its extremal value | J | = 1 / ( 6 3 ) {\displaystyle |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.
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