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

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

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

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

Alternatively, ${\displaystyle \theta _{12}=\theta _{23}=-45^{\circ }}$ and ${\displaystyle \theta _{13}=\delta _{13}=0}$ can be used, which corresponds to: ^[2]: 5

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

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: ^[5]: 7

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

Bimaximal mixing corresponds to ${\displaystyle \theta =45^{\circ }}$. 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 ${\displaystyle \theta _{13}}$. ^[7]

See also

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

References