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# 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 is compatible with all verified neutrino oscillation experiments to date,[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 standard (PDG[2]) convention for the PMNS matrix, tribimaximal mixing may be specified in terms of lepton mixing angles as follows:

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

## 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[5] and bimaximal[6] mixing schemes, both now ruled out by data. In tribimaximal mixing,[1] the ν2 neutrino mass eigenstate is said to be "trimaximally mixed" in that it consists of a uniform admixture of νe, νμ and ντ flavour eigenstates, ie. maximal mixing among all three flavour states. The ν3 neutrino mass eigenstate, on the other hand, is "bimaximally mixed" in that it comprises a uniform admixture of only two flavour components, ie. νμ and ντ maximal mixing, with effective decoupling of the νe from the ν3, just as in the original bimaximal scheme.[6]

## Phenomenology

By virtue of the zero ( | Ue3 | 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 $ \simeq 1/3$ in the Sudbury Neutrino Observatory (SNO) and $ \simeq 5/9$ in lower energy solar neutrino experiments (and in long baseline reactor neutrino experiments). The bimaximally mixed ν3 in tribimximal mixing accounts for the factor of two suppression $ \simeq 1/2$ observed for atmospheric muon-neutrinos (and confirmed in long-baseline accelerator experiments). Near-zero νe appearance in a νμ beam is predicted in exact tribimaximal mixing ( | Ue3 | 2 = 0), and future experiments may well rule this out. Further characteristic predictions[1] of tribimaxiaml mixing, eg. for very long baseline νμ and ντ (vacuum) survival probabilities $(P_{\mu \mu}=P_{\tau \tau} \simeq 7/18)$, will be extremely hard to test experimentally.

## History

The name tribimaximal first appeared in the literature in 2002[1] although this specific scheme had been previously published in 1999[7] as a viable alternative to the trimaximal[5] scheme. Tribimaximal mixing is sometimes confused with other mixing schemes, e.g.[8] 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]

## 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.
2. ^ a b c W.M. Yao et al. (2006). "Particle Data Group - 2006 Review of Particle Physics". Journal of Physics G 33 (1). Neutrino mass, mixing, and flavor change.
3. ^ G. Altarelli and F. Feruglio (1998). "Models of neutrino masses from oscillations with maximal mixing". Journal of High Energy Physics 9811: 021.
4. ^ J.D. Bjorken, P.F. Harrison and W. G. Scott (2006). "Simplified unitarity triangles for the lepton sector". Physical Review D 74: 073012.
5. ^ 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.
6. ^ a b V.D. Barger, S. Pakvasa, T.J. Weiler and K. Whisnant (1998). "Bimaximal mixing of three neutrinos". Physics Letters B 437: 107.
7. ^ 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.
8. ^ L. Wolfenstein (1978). "Oscillations Among Three Neutrino Types and CP Violation". Physical Review D 18: 958.