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 $\omega =\exp(i2\pi /3)$ and ${\bar {\omega }}=\exp(-i2\pi /3)$ are the complex cube roots of unity. In the standard PDG^[3] convention, trimaximal mixing corresponds to: $\theta _{12}=\theta _{23}=\pi /4$ , $\theta _{13}=\sin ^{-1}(1/{\sqrt {3}})$ 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.

References

^ ^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 (1): 79–92. arXiv:hep-ph/9904297. Bibcode:1999PhLB..458...79H. doi:10.1016/S0370-2693(99)00438-4. S2CID 16800198.
^ ^a ^b P.F. Harrison, D. H. Perkins and W. G. Scott (1995). "Threefold Maximal Lepton Mixing and the Solar and Atmospheric Neutrino Deficits" (PDF). Physics Letters B. 349 (1–2): 137–144. Bibcode:1995PhLB..349..137H. doi:10.1016/0370-2693(95)00213-5.
^ W.M. Yao; et al. (2006). "Review of Particle Physics: Neutrino mass, mixing, and flavor change" (PDF). Journal of Physics G. 33 (1): 1. arXiv:astro-ph/0601168. Bibcode:2006JPhG...33....1Y. doi:10.1088/0954-3899/33/1/001. S2CID 117958297.
^ 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 (10): 1039–1042. Bibcode:1985PhRvL..55.1039J. doi:10.1103/PhysRevLett.55.1039. PMID 10031712.
^ L. Wolfenstein (1978). "Oscillations Among Three Neutrino Types and CP Violation". Physical Review D. 18 (3): 958–960. Bibcode:1978PhRvD..18..958W. doi:10.1103/PhysRevD.18.958.
^ N. Cabibbo (1978). "Time Reversal Violation in Neutrino Oscillation". Physics Letters B. 32 (3): 333–335. Bibcode:1978PhLB...72..333C. doi:10.1016/0370-2693(78)90132-6.
^ C. Giunti, C. W. Kim and J. D. Kim (1995). "Atmospheric Neutrino Problem in Maximally Mixed Three Generations of Neutrinos". Physics Letters B. 352 (3–4): 357–364. arXiv:hep-ph/9411219. Bibcode:1995PhLB..352..357G. CiteSeerX 10.1.1.347.269. doi:10.1016/0370-2693(95)00424-J. S2CID 15430104.
^ 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 (1–4): 186–196. arXiv:hep-ph/9702243. Bibcode:1997PhLB..396..186H. doi:10.1016/S0370-2693(97)00105-6. S2CID 14447907.
^ P.F. Harrison & W. G. Scott (1994). "Generation Permutation Symmetry and the Quark Mixing Matrix". Physics Letters B. 333 (3–4): 471–475. arXiv:hep-ph/9406351. Bibcode:1994PhLB..333..471H. doi:10.1016/0370-2693(94)90170-8. S2CID 15792208.
^ M. Apollonio; et al. (2003). "Search for Neutrino Oscillations on a Long Baseline at the CHOOZ Nuclear Power Station". European Physical Journal C. 27 (3): 331–374. arXiv:hep-ex/0301017. Bibcode:2003EPJC...27..331A. doi:10.1140/epjc/s2002-01127-9. S2CID 14226312.
^ P.F. Harrison, D. H. Perkins and W. G. Scott (2002). "Tri-Bimaximal Mixing and the Neutrino Oscillation Data". Physics Letters B. 530 (1–4): 167–173. arXiv:hep-ph/0202074. Bibcode:2002PhLB..530..167H. doi:10.1016/S0370-2693(02)01336-9. S2CID 16751525.

[HPS1999-1] 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". Physics Letters B. 458 (1): 79–92. arXiv:hep-ph/9904297. Bibcode:1999PhLB..458...79H. doi:10.1016/S0370-2693(99)00438-4. S2CID 16800198.

[HPS1995-2] P.F. Harrison, D. H. Perkins and W. G. Scott (1995). "Threefold Maximal Lepton Mixing and the Solar and Atmospheric Neutrino Deficits" (PDF). Physics Letters B. 349 (1–2): 137–144. Bibcode:1995PhLB..349..137H. doi:10.1016/0370-2693(95)00213-5.

[PDG-3] W.M. Yao; et al. (2006). "Review of Particle Physics: Neutrino mass, mixing, and flavor change" (PDF). Journal of Physics G. 33 (1): 1. arXiv:astro-ph/0601168. Bibcode:2006JPhG...33....1Y. doi:10.1088/0954-3899/33/1/001. S2CID 117958297.

[JCP-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 (10): 1039–1042. Bibcode:1985PhRvL..55.1039J. doi:10.1103/PhysRevLett.55.1039. PMID 10031712.

[WOLF78-5] L. Wolfenstein (1978). "Oscillations Among Three Neutrino Types and CP Violation". Physical Review D. 18 (3): 958–960. Bibcode:1978PhRvD..18..958W. doi:10.1103/PhysRevD.18.958.

[CAB78-6] N. Cabibbo (1978). "Time Reversal Violation in Neutrino Oscillation". Physics Letters B. 32 (3): 333–335. Bibcode:1978PhLB...72..333C. doi:10.1016/0370-2693(78)90132-6.

[GIUNTI1995-7] C. Giunti, C. W. Kim and J. D. Kim (1995). "Atmospheric Neutrino Problem in Maximally Mixed Three Generations of Neutrinos". Physics Letters B. 352 (3–4): 357–364. arXiv:hep-ph/9411219. Bibcode:1995PhLB..352..357G. CiteSeerX 10.1.1.347.269. doi:10.1016/0370-2693(95)00424-J. S2CID 15430104.

[HPS1997-8] 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 (1–4): 186–196. arXiv:hep-ph/9702243. Bibcode:1997PhLB..396..186H. doi:10.1016/S0370-2693(97)00105-6. S2CID 14447907.

[HS1994-9] P.F. Harrison & W. G. Scott (1994). "Generation Permutation Symmetry and the Quark Mixing Matrix". Physics Letters B. 333 (3–4): 471–475. arXiv:hep-ph/9406351. Bibcode:1994PhLB..333..471H. doi:10.1016/0370-2693(94)90170-8. S2CID 15792208.

[CHOOZ-10] M. Apollonio; et al. (2003). "Search for Neutrino Oscillations on a Long Baseline at the CHOOZ Nuclear Power Station". European Physical Journal C. 27 (3): 331–374. arXiv:hep-ex/0301017. Bibcode:2003EPJC...27..331A. doi:10.1140/epjc/s2002-01127-9. S2CID 14226312.

[HPS2002-11] P.F. Harrison, D. H. Perkins and W. G. Scott (2002). "Tri-Bimaximal Mixing and the Neutrino Oscillation Data". Physics Letters B. 530 (1–4): 167–173. arXiv:hep-ph/0202074. Bibcode:2002PhLB..530..167H. doi:10.1016/S0370-2693(02)01336-9. S2CID 16751525.

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