Physics:Trimaximal mixing
Trimaximal mixing[1] (also known as threefold maximal mixing[2]) refers to the highly symmetric, maximally CP-violating, [math]\displaystyle{ 3 \times 3 }[/math] fermion mixing configuration, characterised by a unitary matrix ([math]\displaystyle{ U }[/math]) having all its elements equal in modulus ([math]\displaystyle{ |U_{ai}|=1/\sqrt{3} }[/math], [math]\displaystyle{ a,i=1,2,3 }[/math]) as may be written, e.g.:
- [math]\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} }[/math]
where [math]\displaystyle{ \omega=\exp(i2\pi/3) }[/math] and [math]\displaystyle{ \bar{\omega}=\exp(-i2\pi/3) }[/math] are the complex cube roots of unity. In the standard PDG[3] convention, trimaximal mixing corresponds to: [math]\displaystyle{ \theta_{12}=\theta_{23}=\pi/4 }[/math], [math]\displaystyle{ \theta_{13}=\sin^{-1}(1/\sqrt{3}) }[/math] and [math]\displaystyle{ \delta=\pi/2 }[/math]. The Jarlskog [math]\displaystyle{ CP }[/math]-violating parameter [math]\displaystyle{ J }[/math][4] takes its extremal value [math]\displaystyle{ |J|=1/(6\sqrt{3}) }[/math].
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
- ↑ 1.0 1.1 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. doi:10.1016/S0370-2693(99)00438-4. Bibcode: 1999PhLB..458...79H.
- ↑ 2.0 2.1 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 (1–2): 137–144. doi:10.1016/0370-2693(95)00213-5. Bibcode: 1995PhLB..349..137H. http://purl.org/net/epubs/manifestation/9352/RAL-TR-1994-125.pdf.
- ↑ W.M. Yao (2006). "Review of Particle Physics: Neutrino mass, mixing, and flavor change". Journal of Physics G 33 (1): 1. doi:10.1088/0954-3899/33/1/001. Bibcode: 2006JPhG...33....1Y. http://pdg.lbl.gov/2006/reviews/numixrpp.pdf.
- ↑ C. Jarlskog (1985). "Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal [math]\displaystyle{ CP }[/math] Non-Conservation". Physical Review Letters 55 (10): 1039–1042. doi:10.1103/PhysRevLett.55.1039. PMID 10031712. Bibcode: 1985PhRvL..55.1039J.
- ↑ L. Wolfenstein (1978). "Oscillations Among Three Neutrino Types and CP Violation". Physical Review D 18 (3): 958–960. doi:10.1103/PhysRevD.18.958. Bibcode: 1978PhRvD..18..958W.
- ↑ N. Cabibbo (1978). "Time Reversal Violation in Neutrino Oscillation". Physics Letters B 32 (3): 333–335. doi:10.1016/0370-2693(78)90132-6. Bibcode: 1978PhLB...72..333C.
- ↑ 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. doi:10.1016/0370-2693(95)00424-J. Bibcode: 1995PhLB..352..357G.
- ↑ 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. doi:10.1016/S0370-2693(97)00105-6. Bibcode: 1997PhLB..396..186H.
- ↑ P.F. Harrison; W. G. Scott (1994). "Generation Permutation Symmetry and the Quark Mixing Matrix". Physics Letters B 333 (3–4): 471–475. doi:10.1016/0370-2693(94)90170-8. Bibcode: 1994PhLB..333..471H.
- ↑ M. Apollonio (2003). "Search for Neutrino Oscillations on a Long Baseline at the CHOOZ Nuclear Power Station". European Physical Journal C 27 (3): 331–374. doi:10.1140/epjc/s2002-01127-9. Bibcode: 2003EPJC...27..331A.
- ↑ 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. doi:10.1016/S0370-2693(02)01336-9. Bibcode: 2002PhLB..530..167H.
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