# 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:

$\begin{bmatrix} |U_{e 1}|^2 & |U_{e 2}|^2 & |U_{e 3}|^2 \\ |U_{\mu 1}|^2 & |U_{\mu 2}|^2 & |U_{\mu 3}|^2 \\ |U_{\tau 1}|^2 & |U_{\tau 2}|^2 & |U_{\tau 3}|^2 \end{bmatrix} = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & 0 \\ \frac{1}{6} & \frac{1}{3} & \frac{1}{2} \\ \frac{1}{6} & \frac{1}{3} & \frac{1}{2} \end{bmatrix}.$

The tribimaximal mixing form was compatible with all verified neutrino oscillation experiments until recently,[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:

$\begin{matrix} \theta_{12}=\sin^{-1} \left({\frac{1}{\sqrt{3}}}\right)\simeq 35.3^{\circ} & \theta_{23}=45^{\circ}\\ \theta_{13}=0 & \delta=0. \end{matrix}$

The above prediction has been falsified experimentally, because θ13 was found to be large.[5] A large 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 $\nu_2$ neutrino mass eigenstate is said to be "trimaximally mixed" in that it consists of a uniform admixture of $\nu_e$, $\nu_{\mu}$ and $\nu_{\tau}$ 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. $\nu_{\mu}$ and $\nu_{\tau}$ maximal mixing, with effective decoupling of the $\nu_e$ from the $\nu_3$, just as in the original bimaximal scheme.[9] [10]

## Phenomenology

By virtue of the zero ($|U_{e3}|^2=0$) 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 $\langle P_{ee}\rangle \simeq 1/3$ in the Sudbury Neutrino Observatory (SNO) and $\langle P_{ee}\rangle \simeq 5/9$ 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 $\langle P_{\mu \mu}\rangle \simeq 1/2$ observed for atmospheric muon-neutrinos (and confirmed in long-baseline accelerator experiments). Near-zero $\nu_e$ appearance in a $\nu_{\mu}$ beam is predicted in exact tribimaximal mixing ($|U_{e3}|^2 = 0$), and future experiments may well rule this out. Further characteristic predictions[1] of tribimaximal mixing, e.g. for very long baseline $\nu_{\mu}$ and $\nu_{\tau}$ (vacuum) survival probabilities $(P_{\mu \mu}=P_{\tau \tau} \simeq 7/18)$, 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]

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

Additional experimental data fixes $\theta=\sin^{-1} \left({\frac{1}{\sqrt{3}}}\right)$. 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. arXiv:hep-ph/0202074. Bibcode:2002PhLB..530..167H. doi:10.1016/S0370-2693(02)01336-9.
2. ^ a b c
3. ^ G. Altarelli and 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.
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. arXiv:hep-ph/0511201. Bibcode:2006PhRvD..74g3012B. doi:10.1103/PhysRevD.74.073012.
5. ^ F. P. An et al. (Daya Bay Collaboration) (2012). "Observation of electron-antineutrino disappearance at Daya Bay". Physics Review Letters 108 (17): 171803. arXiv:1203.1669. Bibcode:2012PhRvL.108q1803A. doi:10.1103/PhysRevLett.108.171803.
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. arXiv:hep-ph/0102236. Bibcode:2001PhLB..508...79V. doi:10.1016/S0370-2693(01)00485-3.
7. ^ F. Vissani (2001). "A Statistical Approach to Leptonic Mixings and Neutrino Masses". arXiv:hep-ph/0111373 [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. arXiv:hep-ph/9806387. Bibcode:1998PhLB..437..107B. 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. arXiv:hep-ph/9807267. Bibcode:1998MPLA...13.2249A. doi:10.1142/S0217732398002400.
11. ^ I. Stancu and 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.
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. arXiv:hep-ph/0008303. Bibcode:2002MPLA...17...13A. 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. arXiv:hep-ph/9904297. Bibcode:1999PhLB..458...79H. 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.