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,^[3][4] for example those 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 ${\displaystyle \nu _{2}}$ 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.^{[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 ${\displaystyle \nu _{3}}$ 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:^[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[]