Trimaximal mixing explained

Trimaximal mixing^[1] (also known as threefold maximal mixing^[2]) refers to the highly symmetric, maximally CP-violating,

3 x 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} (1)/(\sqrt{ 3) } & \frac & \frac \\\frac & \frac & \frac \\ \frac & \frac & \frac \end\Rightarrow (|U_|^2)=\begin\frac & \frac & \frac \\\frac & \frac & \frac \\ \frac & \frac & \frac \end

where

\omega = \exp(i 2 \pi/ 3 ) and \bar{\omega} = \exp( - i 2 \pi/ 3 ) are the complex cube roots of unity. In the standard PDG^[3] 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 ^[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 no longer tenable (related) tribimaximal mixing^[11] scheme.

Notes and References

1. 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 . . 458 . 1 . 79–92 . hep-ph/9904297 . 1999PhLB..458...79H . 10.1016/S0370-2693(99)00438-4 . 16800198 .
2. P.F. Harrison, D. H. Perkins and W. G. Scott . 1995 . Threefold Maximal Lepton Mixing and the Solar and Atmospheric Neutrino Deficits . . 349 . 1–2 . 137–144 . 1995PhLB..349..137H . 10.1016/0370-2693(95)00213-5 .
3. W.M. Yao. 2006 . Review of Particle Physics: Neutrino mass, mixing, and flavor change . . 33 . 1 . 1. astro-ph/0601168 . 2006JPhG...33....1Y . 10.1088/0954-3899/33/1/001 . 117958297 . etal.
4. C. Jarlskog . 1985 . Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Non-Conservation . . 55 . 10 . 1039–1042 . 1985PhRvL..55.1039J . 10.1103/PhysRevLett.55.1039 . 10031712.
5. L. Wolfenstein . 1978 . Oscillations Among Three Neutrino Types and CP Violation . . 18 . 3 . 958–960 . 1978PhRvD..18..958W . 10.1103/PhysRevD.18.958.
6. N. Cabibbo . 1978 . Time Reversal Violation in Neutrino Oscillation . . 32 . 3 . 333–335 . 1978PhLB...72..333C . 10.1016/0370-2693(78)90132-6 .
7. C. Giunti, C. W. Kim and J. D. Kim . 1995 . Atmospheric Neutrino Problem in Maximally Mixed Three Generations of Neutrinos . . 352 . 3–4 . 357–364 . hep-ph/9411219 . 1995PhLB..352..357G . 10.1016/0370-2693(95)00424-J. 10.1.1.347.269 . 15430104 .
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 . . 396 . 1–4 . 186–196 . hep-ph/9702243 . 1997PhLB..396..186H . 10.1016/S0370-2693(97)00105-6. 14447907 .
9. P.F. Harrison . W. G. Scott . amp . 1994 . Generation Permutation Symmetry and the Quark Mixing Matrix . . 333 . 3–4 . 471–475 . hep-ph/9406351 . 1994PhLB..333..471H . 10.1016/0370-2693(94)90170-8. 15792208 .
10. M. Apollonio. 2003 . Search for Neutrino Oscillations on a Long Baseline at the CHOOZ Nuclear Power Station . . 27 . 3 . 331–374 . hep-ex/0301017 . 2003EPJC...27..331A . 10.1140/epjc/s2002-01127-9. 14226312 . etal.
11. P.F. Harrison, D. H. Perkins and W. G. Scott . 2002 . Tri-Bimaximal Mixing and the Neutrino Oscillation Data . . 530 . 1–4 . 167–173 . hep-ph/0202074 . 2002PhLB..530..167H . 10.1016/S0370-2693(02)01336-9. 16751525 .