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:
\( {\displaystyle {\begin{bmatrix}|U_{e1}|^{2}&|U_{e2}|^{2}&|U_{e3}|^{2}\\|U_{\mu 1}|^{2}&|U_{\mu 2}|^{2}&|U_{\mu 3}|^{2}\\|U_{\tau 1}|^{2}&|U_{\tau 2}|^{2}&|U_{\tau 3}|^{2}\end{bmatrix}}={\begin{bmatrix}{\frac {2}{3}}&{\frac {1}{3}}&0\\{\frac {1}{6}}&{\frac {1}{3}}&{\frac {1}{2}}\\{\frac {1}{6}}&{\frac {1}{3}}&{\frac {1}{2}}\end{bmatrix}}.} \)
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:
\( {\displaystyle {\begin{matrix}\theta _{12}=\sin ^{-1}\left({\frac {1}{\sqrt {3}}}\right)\simeq 35.3^{\circ }&\theta _{23}=45^{\circ }\\\theta _{13}=0&\delta =0.\end{matrix}}} \)
The above prediction has been falsified experimentally, because θ13 was found to be nontrivial, θ13 =8.5°.[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
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][10] mixing schemes, both now ruled out by data. In tribimaximal mixing,[1] the \( \nu_2 \) neutrino mass eigenstate is said to be "trimaximally mixed" in that it consists of a uniform admixture of \( \nu _{e} \), \( {\displaystyle \nu _{\mu }} \) and \( {\displaystyle \nu _{\tau }} \) flavour eigenstates, i.e. maximal mixing among all three flavour states. The \( \nu_3 neutrino \) mass eigenstate, on the other hand, is "bimaximally mixed" in that it comprises a uniform admixture of only two flavour components, i.e. \( {\displaystyle \nu _{\mu }} \) and \( {\displaystyle \nu _{\tau }} \) maximal mixing, with effective decoupling of the \( \nu _{e} \) from the \( \nu_3 \), just as in the original bimaximal scheme.[10] [11]
Phenomenology
By virtue of the zero ( \( {\displaystyle |U_{e3}|^{2}=0} \) ) 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 \( {\displaystyle \langle P_{ee}\rangle \simeq 1/3} \)in the Sudbury Neutrino Observatory (SNO) and \( {\displaystyle \langle P_{ee}\rangle \simeq 5/9} \) in lower energy solar neutrino experiments (and in long baseline reactor neutrino experiments). The bimaximally mixed \( \nu_3 \) in tribimaximal mixing accounts for the factor of two suppression \( {\displaystyle \langle P_{\mu \mu }\rangle \simeq 1/2} \) observed for atmospheric muon-neutrinos (and confirmed in long-baseline accelerator experiments). Near-zero \( {\displaystyle \nu _{e}} \) appearance in a \( {\displaystyle \nu _{\mu }} \) beam is predicted in exact tribimaximal mixing ( \( {\displaystyle |U_{e3}|^{2}=0}) \) , and future experiments may well rule this out. Further characteristic predictions[1] of tribimaximal mixing, e.g. for very long baseline ν μ {\displaystyle \nu _{\mu }} {\displaystyle \nu _{\mu }} and \( {\displaystyle \nu _{\tau }} \) (vacuum) survival probabilities ( \( {\displaystyle (P_{\mu \mu }=P_{\tau \tau }\simeq 7/18)} \), 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:[12]
\( {\displaystyle U={\begin{bmatrix}\cos \theta &\sin \theta &0\\-\sin \theta /{\sqrt {2}}&\cos \theta /{\sqrt {2}}&{\frac {1}{\sqrt {2}}}\\\sin \theta /{\sqrt {2}}&-\cos \theta /{\sqrt {2}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}}.} \)
Additional experimental data fixes \( {\displaystyle \theta =\sin ^{-1}\left({\frac {1}{\sqrt {3}}}\right)} \). The extension of this result to the CP violating case is found in.[13]
History
The name tribimaximal first appeared in the literature in 2002[1] although this specific scheme had been previously published in 1999[14] as a viable alternative to the trimaximal[8] scheme. Tribimaximal mixing is sometimes confused with other mixing schemes, e.g.[15] 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.
References
P. F. Harrison; D. H. Perkins; W. G. Scott (2002). "Tribimaximal 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.
W. M. Yao; et al. (Particle Data Group) (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.
G. Altarelli & F. Feruglio (1998). "Models of neutrino masses from oscillations with maximal mixing". Journal of High Energy Physics. 1998 (11): 021. arXiv:hep-ph/9809596. Bibcode:1998JHEP...11..021A. doi:10.1088/1126-6708/1998/11/021. S2CID 15333617.
J. D. Bjorken; P. F. Harrison; W. G. Scott (2006). "Simplified unitarity triangles for the lepton sector". Physical Review D (Submitted manuscript). 74 (7): 073012. arXiv:hep-ph/0511201. Bibcode:2006PhRvD..74g3012B. doi:10.1103/PhysRevD.74.073012. S2CID 5353114.
Patrignani, C.; et al. (Particle Data Group) (2016). Updated June 2016 by K. Nakamura & S.T. Petcov. "Neutrino Mass, Mixing, and Oscillations" (PDF). Chin. Phys. C. 40: 100001. Archived from the original (PDF) on 2017-11-15.
F. Vissani (2001). "Expected properties of massive neutrinos for mass matrices with a dominant block and random coefficients order unity". Physics Letters B. 508 (1–2): 79–84. arXiv:hep-ph/0102236. Bibcode:2001PhLB..508...79V. CiteSeerX 10.1.1.346.1568. doi:10.1016/S0370-2693(01)00485-3. S2CID 2637568.
F. Vissani (2001). "A Statistical Approach to Leptonic Mixings and Neutrino Masses". arXiv:hep-ph/0111373.
P. F. Harrison; D. H. Perkins; W. G. Scott (1995). "Threefold maximal lepton mixing and the solar and atmospheric neutrino deficits". Physics Letters B. 349 (1–2): 137–144. Bibcode:1995PhLB..349..137H. doi:10.1016/0370-2693(95)00213-5.
F. Vissani (1997). "A study of the scenario with nearly degenerate Majorana neutrinos". arXiv:hep-ph/9708483.
V. D. Barger; S. Pakvasa; T. J. Weiler; K. Whisnant (1998). "Bimaximal mixing of three neutrinos". Physics Letters B. 437 (1–2): 107–116. arXiv:hep-ph/9806387. Bibcode:1998PhLB..437..107B. CiteSeerX 10.1.1.345.3379. doi:10.1016/S0370-2693(98)00880-6. S2CID 14622000.
D. V. Ahluwalia (1998). "On Reconciling Atmospheric, LSND, and Solar Neutrino-Oscillation Data". Modern Physics Letters A. 13 (28): 2249–2264. arXiv:hep-ph/9807267. Bibcode:1998MPLA...13.2249A. doi:10.1142/S0217732398002400. S2CID 18101181.
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. arXiv:hep-ph/9903408. Bibcode:1999PhLB..460..431S. doi:10.1016/S0370-2693(99)00811-4. S2CID 14787873.
D. V. Ahluwalia; Y. Liu; 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 (1): 13–21. arXiv:hep-ph/0008303. Bibcode:2002MPLA...17...13A. doi:10.1142/S0217732302006138. S2CID 18910986.
P. F. Harrison; D. H. Perkins; 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.
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