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:

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

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:

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

1 Explanation of name
2 Phenomenology
3 History
4 References

[edit] 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 $ν 2$ neutrino mass eigenstate is said to be "trimaximally mixed" in that it consists of a uniform admixture of $ν e$ , $ν μ$ and $ν τ$ flavour eigenstates, ie. maximal mixing among all three flavour states. The $ν 3$ neutrino mass eigenstate, on the other hand, is "bimaximally mixed" in that it comprises a uniform admixture of only two flavour components, ie. $ν μ$ and $ν τ$ maximal mixing, with effective decoupling of the $ν e$ from the $ν 3$ , just as in the original bimaximal scheme.^[6]

[edit] Phenomenology

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

[edit] 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]