Bimaximal mixing

Bimaximal mixing refers to a proposed form of the lepton mixing matrix. It is characterized by the $\nu _{3}$ neutrino being a bimaximal mixture of $\nu _{\mu }$ and $\nu _{\tau }$ and being completely decoupled from the $\nu _{e}$, i.e. a uniform mixture of $\nu _{\mu }$ and $\nu _{\tau }$. The $\nu _{e}$ is consequently a uniform mixture of $\nu _{1}$ and $\nu _{2}$. Other notable properties are the symmetries between the $\nu _{\mu }$ and $\nu _{\tau }$ flavours and $\nu _{1}$ and $\nu _{2}$ mass eigenstates and an absence of CP violation. The moduli squared of the matrix elements have to be:

${\begin{bmatrix}|U_{e1}|^{2}&|U_{e2}|^{2}&|U_{e3}|^{2}\\|U_{\mu 1}|^{2}&|U_{\mu 2}|^{2}&|U_{\mu 3}|^{2}\\|U_{\tau 1}|^{2}&|U_{\tau 2}|^{2}&|U_{\tau 3}|^{2}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2}}&0\\{\frac {1}{4}}&{\frac {1}{4}}&{\frac {1}{2}}\\{\frac {1}{4}}&{\frac {1}{4}}&{\frac {1}{2}}\end{bmatrix}}$.

According to PDG convention, bimaximal mixing corresponds to $\theta _{12}=\theta _{23}=45^{\circ }$ and $\theta _{13}=\delta _{13}=0$, which produces following matrix:

${\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\-{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{\sqrt {2}}}\\{\frac {1}{2}}&-{\frac {1}{2}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}}$.

Alternatively, $\theta _{12}=\theta _{23}=-45^{\circ }$ and $\theta _{13}=\delta _{13}=0$ can be used, which corresponds to:

${\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{\sqrt {2}}}&-{\frac {1}{\sqrt {2}}}&0\\{\frac {1}{2}}&{\frac {1}{2}}&-{\frac {1}{\sqrt {2}}}\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}}$.

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:

$U={\begin{bmatrix}\cos \theta &\sin \theta &0\\-\sin \theta /{\sqrt {2}}&\cos \theta /{\sqrt {2}}&{\frac {1}{\sqrt {2}}}\\\sin \theta /{\sqrt {2}}&-\cos \theta /{\sqrt {2}}&{\frac {1}{\sqrt {2}}}\end{bmatrix}}.$

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

See also

• Trimaximal mixing
• Tribimaximal mixing
• Neutrino oscillation
• Double Chooz