Bimaximal mixing

Bimaximal mixing refers to a proposed form of the lepton mixing matrix.[1][2] It is characterized by the [math]\displaystyle{ \nu_3 }[/math] neutrino being a bimaximal mixture of [math]\displaystyle{ \nu_\mu }[/math] and [math]\displaystyle{ \nu_\tau }[/math] and being completely decoupled from the [math]\displaystyle{ \nu_e }[/math], i.e. a uniform mixture of [math]\displaystyle{ \nu_\mu }[/math] and [math]\displaystyle{ \nu_\tau }[/math]. The [math]\displaystyle{ \nu_e }[/math] is consequently a uniform mixture of [math]\displaystyle{ \nu_1 }[/math] and [math]\displaystyle{ \nu_2 }[/math]. Other notable properties are the symmetries between the [math]\displaystyle{ \nu_\mu }[/math] and [math]\displaystyle{ \nu_\tau }[/math] flavours and [math]\displaystyle{ \nu_1 }[/math] and [math]\displaystyle{ \nu_2 }[/math] mass eigenstates and an absence of CP violation. The moduli squared of the matrix elements have to be:

[math]\displaystyle{ \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{1}{2} & \frac{1}{2} & 0 \\\ \frac{1}{4} & \frac{1}{4} & \frac{1}{2} \\\ \frac{1}{4} & \frac{1}{4} & \frac{1}{2} \end{bmatrix} }[/math].

According to PDG convention[3]:7, bimaximal mixing corresponds to [math]\displaystyle{ \theta_{12}=\theta_{23}=45^\circ }[/math] and [math]\displaystyle{ \theta_{13}=\delta_{13}=0 }[/math], which produces following matrix:[4]:24

[math]\displaystyle{ \begin{bmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}} \\\ \frac{1}{2} & -\frac{1}{2} & \frac{1}{\sqrt{2}} \end{bmatrix} }[/math].

Alternatively, [math]\displaystyle{ \theta_{12}=\theta_{23}=-45^\circ }[/math] and [math]\displaystyle{ \theta_{13}=\delta_{13}=0 }[/math] can be used, which corresponds to:[2]:5

[math]\displaystyle{ \begin{bmatrix} U_{e 1} & U_{e 2} & U_{e 3} \\\ U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\\ U_{\tau 1} & U_{\tau 2} & U_{\tau 3} \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \\\ \frac{1}{2} & \frac{1}{2} & -\frac{1}{\sqrt{2}} \\\ \frac{1}{2} & \frac{1}{2} & \frac{1}{\sqrt{2}} \end{bmatrix} }[/math].

## Phenomenology

The L/E flatness of the electron-like event ratio at Super-Kamiokande severely restricts the CP-conserving neutrino mixing matrices to the form:[5]:7

[math]\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}. }[/math]

Bimaximal mixing corresponds to [math]\displaystyle{ \theta=45^\circ }[/math]. Tribimaximal mixing and golden-ratio mixing also correspond to an angle in the above parametrization.[6] Bimaximal mixing, along with these other mixing schemes, have been falsified by a non-zero [math]\displaystyle{ \theta_{13} }[/math].[7]

## See also

* Trimaximal mixing
* Tribimaximal mixing
* Neutrino oscillation
* Double Chooz

## References

1. ↑ F. Vissani (1997). A study of the scenario with nearly degenerate Majorana neutrinos. Bibcode: 1997hep.ph....8483V.
2. ↑ 2.0 2.1 V. D. Barger; S. Pakvasa; T. J. Weiler; K. Whisnant (1998). "Bimaximal mixing of three neutrinos". Physics Letters B 437 (1–2): 107–116. doi:10.1016/S0370-2693(98)00880-6. Bibcode: 1998PhLB..437..107B.
3. ↑ Gonzalez-Garcia, M.C.; Yokoyama, M. (August 2019). "14. Neutrino Masses, Mixing, and Oscillations". https://pdg.lbl.gov/2021/reviews/rpp2020-rev-neutrino-mixing.pdf.
4. ↑ King, Steve (August 2014). "Neutrino Mass Models - Lecture 1: Lepton Mixing". https://indico.cern.ch/event/300715/contributions/686782/attachments/566732/780663/king_lecture1.pdf.
5. ↑ 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. doi:10.1016/S0370-2693(99)00811-4. Bibcode: 1999PhLB..460..431S.
6. ↑ Zhang, Jue; Zhou, Shun (July 25, 2016). "Viability of exact tri-bimaximal, golden-ratio and bimaximal mixing patterns and renormalization-group running effects". Journal of High Energy Physics 2016 (167): 167. doi:10.1007/JHEP09(2016)167. Bibcode: 2016JHEP...09..167Z.
7. ↑ Abe, Y. (2014). "Improved measurements of the neutrino mixing angle θ13 with the Double Chooz detector". Journal of High Energy Physics 2014 (10): 86. doi:10.1007/JHEP10(2014)086. Bibcode: 2014JHEP...10..086A.

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