# Tribimaximal mixing

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 =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

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 ${\displaystyle \nu _{2}}$ neutrino mass eigenstate is said to be "trimaximally mixed" in that it consists of a uniform admixture of ${\displaystyle \nu _{e}}$, ${\displaystyle \nu _{\mu }}$ and ${\displaystyle \nu _{\tau }}$ flavour eigenstates, i.e. maximal mixing among all three flavour states. The ${\displaystyle \nu _{3}}$ 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 ${\displaystyle \nu _{e}}$ from the ${\displaystyle \nu _{3}}$, just as in the original bimaximal scheme.[9] [10]

## 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 ${\displaystyle \nu _{3}}$ 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 ${\displaystyle \nu _{\mu }}$ 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:[11]

${\displaystyle U={\begin{bmatrix}\cos \theta &\sin \theta &0\\-\sin \theta /{\sqrt {2}}&\cos \theta /{\sqrt {2}}&{\frac {1}{\sqrt {2}}}\\\sin \theta /{\sqrt {2}}&-\cos \theta /{\sqrt {2}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}}.}$

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.[12]

## History

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.

## References

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:. 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:. 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:. 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:. 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:. doi:10.1016/S0370-2693(01)00485-3.
7. ^ F. Vissani (2001). "A Statistical Approach to Leptonic Mixings and Neutrino Masses". arXiv: [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:. 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:. 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:. 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:. 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:. 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.