Bimaximal mixing refers to a proposed form of the lepton mixing matrix .[ 1] [ 2] It is characterized by the
ν ν -->
3
{\displaystyle \nu _{3}}
neutrino being a bimaximal mixture of
ν ν -->
μ μ -->
{\displaystyle \nu _{\mu }}
and
ν ν -->
τ τ -->
{\displaystyle \nu _{\tau }}
and being completely decoupled from the
ν ν -->
e
{\displaystyle \nu _{e}}
, i.e. a uniform mixture of
ν ν -->
μ μ -->
{\displaystyle \nu _{\mu }}
and
ν ν -->
τ τ -->
{\displaystyle \nu _{\tau }}
. The
ν ν -->
e
{\displaystyle \nu _{e}}
is consequently a uniform mixture of
ν ν -->
1
{\displaystyle \nu _{1}}
and
ν ν -->
2
{\displaystyle \nu _{2}}
. Other notable properties are the symmetries between the
ν ν -->
μ μ -->
{\displaystyle \nu _{\mu }}
and
ν ν -->
τ τ -->
{\displaystyle \nu _{\tau }}
flavours and
ν ν -->
1
{\displaystyle \nu _{1}}
and
ν ν -->
2
{\displaystyle \nu _{2}}
mass eigenstates and an absence of CP violation . The moduli squared of the matrix elements have to be:
[
|
U
e
1
|
2
|
U
e
2
|
2
|
U
e
3
|
2
|
U
μ μ -->
1
|
2
|
U
μ μ -->
2
|
2
|
U
μ μ -->
3
|
2
|
U
τ τ -->
1
|
2
|
U
τ τ -->
2
|
2
|
U
τ τ -->
3
|
2
]
=
[
1
2
1
2
0
1
4
1
4
1
2
1
4
1
4
1
2
]
{\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
θ θ -->
12
=
θ θ -->
23
=
45
∘ ∘ -->
{\displaystyle \theta _{12}=\theta _{23}=45^{\circ }}
and
θ θ -->
13
=
δ δ -->
13
=
0
{\displaystyle \theta _{13}=\delta _{13}=0}
, which produces following matrix:[ 4] : 24
[
U
e
1
U
e
2
U
e
3
U
μ μ -->
1
U
μ μ -->
2
U
μ μ -->
3
U
τ τ -->
1
U
τ τ -->
2
U
τ τ -->
3
]
=
[
1
2
1
2
0
− − -->
1
2
1
2
1
2
1
2
− − -->
1
2
1
2
]
{\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,
θ θ -->
12
=
θ θ -->
23
=
− − -->
45
∘ ∘ -->
{\displaystyle \theta _{12}=\theta _{23}=-45^{\circ }}
and
θ θ -->
13
=
δ δ -->
13
=
0
{\displaystyle \theta _{13}=\delta _{13}=0}
can be used, which corresponds to:[ 2] : 5
[
U
e
1
U
e
2
U
e
3
U
μ μ -->
1
U
μ μ -->
2
U
μ μ -->
3
U
τ τ -->
1
U
τ τ -->
2
U
τ τ -->
3
]
=
[
1
2
− − -->
1
2
0
1
2
1
2
− − -->
1
2
1
2
1
2
1
2
]
{\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
U
=
[
cos
-->
θ θ -->
sin
-->
θ θ -->
0
− − -->
sin
-->
θ θ -->
/
2
cos
-->
θ θ -->
/
2
1
2
sin
-->
θ θ -->
/
2
− − -->
cos
-->
θ θ -->
/
2
1
2
]
.
{\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
θ θ -->
=
45
∘ ∘ -->
{\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
θ θ -->
13
{\displaystyle \theta _{13}}
.[ 7]
See also
References
^
F. Vissani (1997). "A study of the scenario with nearly degenerate Majorana neutrinos". arXiv :hep-ph/9708483 .
^ a b
V. D. Barger; S. Pakvasa; T. J. Weiler; K. Whisnant (1998). "Bimaximal mixing of three neutrinos". Physics Letters B . 437 (1–2): 107–116. arXiv :hep-ph/9806387 . Bibcode :1998PhLB..437..107B . CiteSeerX 10.1.1.345.3379 . doi :10.1016/S0370-2693(98)00880-6 . S2CID 14622000 .
^ Gonzalez-Garcia, M.C.; Yokoyama, M. (August 2019). "14. Neutrino Masses, Mixing, and Oscillations" (PDF) . Retrieved 16 June 2021 .
^ King, Steve (August 2014). "Neutrino Mass Models - Lecture 1: Lepton Mixing" (PDF) . Retrieved 16 June 2021 .
^
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. arXiv :hep-ph/9903408 . Bibcode :1999PhLB..460..431S . doi :10.1016/S0370-2693(99)00811-4 . S2CID 14787873 .
^
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. arXiv :1606.09591 . Bibcode :2016JHEP...09..167Z . doi :10.1007/JHEP09(2016)167 . S2CID 119208235 .
^
Abe, Y.; et al. (Double Chooz Collaboration) (2014). "Improved measurements of the neutrino mixing angle θ13 with the Double Chooz detector". Journal of High Energy Physics . 2014 (10): 86. arXiv :1406.7763 . Bibcode :2014JHEP...10..086A . doi :10.1007/JHEP10(2014)086 . S2CID 53849018 .