Tribimaximal mixing

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Tribimaximal mixing[1] is a specific postulated form for the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) lepton mixing matrix U. Tribimaximal mixing is defined by a particular choice of the matrix of moduli-squared of the elements of the PMNS matrix as follows:

This mixing is presently excluded by experiment at the level of 5σ.

The tribimaximal mixing form was compatible with much older neutrino oscillation experiments [2] and may be used as a zeroth-order approximation to more general forms for the PMNS matrix e.g.[3][4] which are also consistent with the data. In the PDG[2] convention for the PMNS matrix, tribimaximal mixing may be specified in terms of lepton mixing angles as follows:

The above prediction has been falsified experimentally, because θ13 was found to be nontrivial, θ13 =0.15.[5]

A non-negligible value of θ13 has been foreseen in certain theoretical schemes that were put forward before tribimaximal mixing and that supported a large solar mixing, before it was confirmed experimentally [6][7] (these theoretical schemes do not have a special name, but for the reasons explained above, they could be called pre-tribimaximal or also non-tribimaximal). This situation is not new: also in the 1990s, the solar mixing angle was supposed to be small by most theorists, until KamLAND proved the contrary to be true.

Explanation of name[edit]

The name tribimaximal reflects the commonality of the tribimaximal mixing matrix with two previously proposed specific forms for the PMNS matrix, the trimaximal[8] and bimaximal[9] mixing schemes, both now ruled out by data. In tribimaximal mixing,[1] the neutrino mass eigenstate is said to be "trimaximally mixed" in that it consists of a uniform admixture of , and flavour eigenstates, i.e. maximal mixing among all three flavour states. The neutrino mass eigenstate, on the other hand, is "bimaximally mixed" in that it comprises a uniform admixture of only two flavour components, i.e. and maximal mixing, with effective decoupling of the from the , just as in the original bimaximal scheme.[9] [10]


By virtue of the zero () in the tribimaximal mixing matrix, exact tribimaximal mixing would predict zero for all CP-violating asymmetries in the case of Dirac neutrinos (in the case of Majorana neutrinos, Majorana phases are still permitted, and could still lead to CP-violating effects).

For solar neutrinos the large angle MSW effect in tribimaximal mixing accounts for the experimental data, predicting average suppressions in the Sudbury Neutrino Observatory (SNO) and in lower energy solar neutrino experiments (and in long baseline reactor neutrino experiments). The bimaximally mixed in tribimaximal mixing accounts for the factor of two suppression observed for atmospheric muon-neutrinos (and confirmed in long-baseline accelerator experiments). Near-zero appearance in a beam is predicted in exact tribimaximal mixing (), and future experiments may well rule this out. Further characteristic predictions[1] of tribimaximal mixing, e.g. for very long baseline and (vacuum) survival probabilities , will be extremely hard to test experimentally.

The L/E flatness of the electron-like event ratio at Super-Kamiokande severely restricts the neutrino mixing matrices to the form:[11]

Additional experimental data fixes . The extension of this result to the CP violating case is found in.[12]


The name tribimaximal first appeared in the literature in 2002[1] although this specific scheme had been previously published in 1999[13] as a viable alternative to the trimaximal[8] scheme. Tribimaximal mixing is sometimes confused with other mixing schemes, e.g.[14] which differ from tribimaximal mixing by row- and/or column-wise permutations of the mixing-matrix elements. Such permuted forms are experimentally distinct however, and are now ruled out by data.[2]

That the L/E flatness of the electron-like event ratio at Superkamiokande severely restricts the neutrino mixing matrices was first presented by D. V. Ahluwalia in a Nuclear and Particle Physics Seminar of the Los Alamos National Laboratory on June 5, 1998. It was just a few hours after the Super-Kamiokande press conference that announced the results on atmospheric neutrinos.


  1. ^ a b c d P. F. Harrison, D. H. Perkins and W. G. Scott (2002). "Tribimaximal mixing and the neutrino oscillation data". Physics Letters B. 530: 167. Bibcode:2002PhLB..530..167H. arXiv:hep-ph/0202074Freely accessible. doi:10.1016/S0370-2693(02)01336-9. 
  2. ^ a b c W. M. Yao; Particle Data Group; et al. (2006). "Review of Particle Physics: Neutrino mass, mixing, and flavor change" (PDF). Journal of Physics G. 33: 1. Bibcode:2006JPhG...33....1Y. arXiv:astro-ph/0601168Freely accessible. doi:10.1088/0954-3899/33/1/001. 
  3. ^ G. Altarelli & F. Feruglio (1998). "Models of neutrino masses from oscillations with maximal mixing". Journal of High Energy Physics. 1998 (11): 021. Bibcode:1998JHEP...11..021A. arXiv:hep-ph/9809596Freely accessible. doi:10.1088/1126-6708/1998/11/021. 
  4. ^ J. D. Bjorken, P. F. Harrison and W. G. Scott (2006). "Simplified unitarity triangles for the lepton sector". Physical Review D. 74 (7): 073012. Bibcode:2006PhRvD..74g3012B. arXiv:hep-ph/0511201Freely accessible. doi:10.1103/PhysRevD.74.073012. 
  5. ^ Nakamura & Petrov, PDG 2017
  6. ^ F. Vissani (2001). "Expected properties of massive neutrinos for mass matrices with a dominant block and random coefficients order unity". Physics Letters B. 508: 79. Bibcode:2001PhLB..508...79V. arXiv:hep-ph/0102236Freely accessible. doi:10.1016/S0370-2693(01)00485-3. 
  7. ^ F. Vissani (2001). "A Statistical Approach to Leptonic Mixings and Neutrino Masses". arXiv:hep-ph/0111373Freely accessible [hep-ph]. 
  8. ^ 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. Bibcode:1995PhLB..349..137H. doi:10.1016/0370-2693(95)00213-5. 
  9. ^ a b V. D. Barger, S. Pakvasa, T. J. Weiler and K. Whisnant (1998). "Bimaximal mixing of three neutrinos". Physics Letters B. 437: 107. Bibcode:1998PhLB..437..107B. arXiv:hep-ph/9806387Freely accessible. doi:10.1016/S0370-2693(98)00880-6. 
  10. ^ D. V. Ahluwalia (1998). "On Reconciling Atmospheric, LSND, and Solar Neutrino-Oscillation Data". Modern Physics Letters A. 13 (28): 2249–2264. Bibcode:1998MPLA...13.2249A. arXiv:hep-ph/9807267Freely accessible. doi:10.1142/S0217732398002400. 
  11. ^ I. Stancu & D. V. Ahluwalia (1999). "L/E-Flatness of the Electron-Like Event Ratio in Super-Kamiokande and a Degeneracy in Neutrino Masses". Physics Letters B. 460 (3–4): 431–436. Bibcode:1999PhLB..460..431S. arXiv:hep-ph/9903408Freely accessible. doi:10.1016/S0370-2693(99)00811-4. 
  12. ^ D. V. Ahluwalia, Y. Liu and I. Stancu (2002). "CP-Violation in Neutrino Oscillations and L/E Flatness of the E-like Event Ratio at Super-Kamiokande". Modern Physics Letters A. 17: 13–21. Bibcode:2002MPLA...17...13A. arXiv:hep-ph/0008303Freely accessible. doi:10.1142/S0217732302006138. 
  13. ^ 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. Bibcode:1999PhLB..458...79H. arXiv:hep-ph/9904297Freely accessible. doi:10.1016/S0370-2693(99)00438-4. 
  14. ^ L. Wolfenstein (1978). "Oscillations Among Three Neutrino Types and CP Violation". Physical Review D. 18 (3): 958. Bibcode:1978PhRvD..18..958W. doi:10.1103/PhysRevD.18.958.