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# Trimaximal mixing

Trimaximal mixing [1] (also known as threefold maximal mixing [2]) 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 ω = exp(i2π / 3) and $\bar{\omega}=\exp(-i2\pi/3)$ are the complex cube roots of unity. In the standard PDG [3] convention, trimaximal mixing corresponds to: θ12 = θ23 = π / 4, $\theta_{13}=\sin^{-1}(1/\sqrt{3})$ and δ = π / 2. The Jarlskog CP-violating parameter J [4] takes its extremal value $|J|=1/(6\sqrt{3})$.

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 (related) tribimaximal mixing [11] scheme.

## References

1. ^ a b P.F. Harrison, D. H. Perkins and W. G. Scott (1999). "A Redetermination of the Neutrino Mass-Squared Difference in Tri-Maximal Mixing with Terrestrial Matter Effects". Physics Letters B 458: 79.
2. ^ a b P.F. Harrison, D. H. Perkins and W. G. Scott (1995). "Threefold Maximal Lepton Mixing and the Solar and Atmospheric Neutrino Deficits". Physics Letters B 349: 137.
3. ^ W.M. Yao et al. (2006). "Particle Data Group - 2006 Review of Particle Physics". Journal of Physics G 33 (1). Neutrino mass, mixing, and flavor change.
4. ^ C. Jarlskog (1985). "Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Non-Conservation". Physical Review Letters 55: 1039.
5. ^ L. Wolfenstein (1978). "Oscillations Among Three Neutrino Types and CP Violation". Physical Review D 18: 958.
6. ^ N. Cabibbo (1978). "Time Reversal Violation in Neutrino Oscillation". Physics Letters B 32: 333.
7. ^ C. Giunti, C. W. Kim and J. D. Kim, (1995). "Atmospheric Neutrino Problem in Maximally Mixed Three Generations of Neutrinos". Physics Letters B 352: 357.
8. ^ P.F. Harrison, D. H. Perkins and W. G. Scott (1997). "Further Evidence for Threefold Maximal Lepton Mixing and a Hierarchical Spectrum of Neutrino Mass-Squared Differences". Physics Letters B 396: 186.
9. ^ P.F. Harrison and W. G. Scott (1994). "Generation Permutation Symmetry and the Quark Mixing Matrix". Physics Letters B 333: 471.
10. ^ M. Apollonio et al. (2003). "Search for Neutrino Oscillations on a Long Baseline at the CHOOZ Nuclear Power Station". European Journal of Physics J C27: 331.
11. ^ P.F. Harrison, D. H. Perkins and W. G. Scott (2002). "Tri-Bimaximal Mixing and the Neutrino Oscillation Data". Physics Letters B 530: 167.