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

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