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:

$\left[\begin{array}{ccc}|U_{e1}{|}^2& |U_{e2}{|}^2& |U_{e3}{|}^2\\ |U_{\mu 1}{|}^2& |U_{\mu 2}{|}^2& |U_{\mu 3}{|}^2\\ |U_{\tau 1}{|}^2& |U_{\tau 2}{|}^2& |U_{\tau 3}{|}^2\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{2}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{2}\end{array}\right]$.

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

$\left[\begin{array}{ccc}{U}_{e1}& U_{e2}& U_{e3}\\ U_{\mu 1}& U_{\mu 2}& U_{\mu 3}\\ U_{\tau 1}& U_{\tau 2}& U_{\tau 3}\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\ -\frac{1}{2}& \frac{1}{2}& \frac{1}{\sqrt{2}}\\ \frac{1}{2}& -\frac{1}{2}& \frac{1}{\sqrt{2}}\end{array}\right]$.

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

$\left[\begin{array}{ccc}{U}_{e1}& U_{e2}& U_{e3}\\ U_{\mu 1}& U_{\mu 2}& U_{\mu 3}\\ U_{\tau 1}& U_{\tau 2}& U_{\tau 3}\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}& 0\\ \frac{1}{2}& \frac{1}{2}& -\frac{1}{\sqrt{2}}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{\sqrt{2}}\end{array}\right]$.

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:Template:R

$U=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0\\ -\sin \theta /\sqrt{2}& \cos \theta /\sqrt{2}& \frac{1}{\sqrt{2}}\\ \sin \theta /\sqrt{2}& -\cos \theta /\sqrt{2}& \frac{1}{\sqrt{2}}\end{array}\right].$

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