Tribimaximal mixing
Encyclopedia
Tribimaximal mixing is a specific postulated form for the Pontecorvo–Maki–Nakagawa–Sakata
(PMNS) lepton mixing matrix U. Tribimaximal mixing is defined by a particular choice of the matrix of moduli-squared of the elements of the PMNS matrix as follows:
The tribimaximal mixing form is compatible with all verified neutrino oscillation
experiments to date, and may be used as a zeroth-order approximation
to more general forms for the PMNS matrix e.g. which are also consistent with the data. In the standard (PDG) convention for the PMNS matrix, tribimaximal mixing may be specified in terms of lepton mixing angles
as follows:
and bimaximal mixing schemes, both now ruled out by data. In tribimaximal mixing, the
neutrino
mass eigenstate is said to be "trimaximally mixed" in that it consists of a uniform admixture of
, 
and 
flavour
eigenstates, i.e. maximal mixing among all three flavour states. The
neutrino mass eigenstate, on the other hand, is "bimaximally mixed" in that it comprises a uniform admixture of only two flavour components, i.e. 
and 
maximal mixing, with effective decoupling of the 
from the 
, just as in the original bimaximal scheme
.
) in the tribimaximal mixing matrix, exact tribimaximal mixing would predict zero for all CP-violating
asymmetries in the case of Dirac neutrinos
(in the case of Majorana neutrinos
, Majorana phases are still permitted, and could still lead to CP-violating effects).
For solar neutrinos
the large angle MSW effect in tribimaximal mixing accounts for the experimental data
, predicting average suppressions
in the Sudbury Neutrino Observatory
(SNO) and
in lower energy solar neutrino experiments (and in long baseline reactor neutrino experiments). The bimaximally mixed 
in tribimaximal mixing accounts for the factor of two suppression 
observed for atmospheric muon-neutrinos
(and confirmed in long-baseline accelerator experiments). Near-zero
appearance in a 
beam is predicted in exact tribimaximal mixing (
), and future experiments
may well rule this out. Further characteristic predictions of tribimaximal mixing, e.g. for very long baseline
and 
(vacuum) survival probabilities 
, will be extremely hard to test experimentally.
The L/E flatness of the electron-like event ratio at Super-Kamiokande
severely restricts the neutrino mixing matrices
to the form
Additional experimental data fixes
.
The extension of this result to the CP violating case is found in reference .
scheme. Tribimaximal mixing is sometimes confused with other mixing schemes, e.g. which differ from tribimaximal mixing by row- and/or column-wise permutations of the mixing-matrix elements. Such permuted forms are experimentally distinct however, and are now ruled out by data.
That the L/E flatness of the electron-like event ratio at Superkamiokande severely restricts the neutrino mixing matrices was first presented by D. V. Ahluwalia in a Nuclear and Particle Physics Seminar of the Los Alamos National Laboratory
on June 5, 1998. It was just a few hours after the Super-Kamiokande
press conference that announced the results on atmospheric neutrinos.
Pontecorvo–Maki–Nakagawa–Sakata matrix
In particle physics, the Pontecorvo–Maki–Nakagawa–Sakata matrix , Maki–Nakagawa–Sakata matrix , lepton mixing matrix, or neutrino mixing matrix, is a unitary matrixThe PMNS matrix is not unitary in the seesaw model which contains information on the mismatch of quantum states of leptons when they...
(PMNS) lepton mixing matrix U. Tribimaximal mixing is defined by a particular choice of the matrix of moduli-squared of the elements of the PMNS matrix as follows:
The tribimaximal mixing form is compatible with all verified neutrino oscillation
Neutrino oscillation
Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvowhereby a neutrino created with a specific lepton flavor can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates...
experiments to date, and may be used as a zeroth-order approximation
Orders of approximation
In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth order approximation, a first order...
to more general forms for the PMNS matrix e.g. which are also consistent with the data. In the standard (PDG) convention for the PMNS matrix, tribimaximal mixing may be specified in terms of lepton mixing angles
Neutrino oscillation
Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvowhereby a neutrino created with a specific lepton flavor can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates...
as follows:
Explanation of name
The name tribimaximal reflects the commonality of the tribimaximal mixing matrix with two previously proposed specific forms for the PMNS matrix, the trimaximalTrimaximal mixing
Trimaximal mixing refers to the highly symmetric, maximally CP-violating, 3 \times 3 fermion mixing configuration, characterised by a unitary matrix having all its elements equal in modulus...
and bimaximal mixing schemes, both now ruled out by data. In tribimaximal mixing, the
neutrinoNeutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
mass eigenstate is said to be "trimaximally mixed" in that it consists of a uniform admixture of
, and flavourFlavour (particle physics)
In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...
eigenstates, i.e. maximal mixing among all three flavour states. The
neutrino mass eigenstate, on the other hand, is "bimaximally mixed" in that it comprises a uniform admixture of only two flavour components, i.e. and maximal mixing, with effective decoupling of the from the , just as in the original bimaximal scheme .
Phenomenology
By virtue of the zero () in the tribimaximal mixing matrix, exact tribimaximal mixing would predict zero for all CP-violatingCP violation
In particle physics, CP violation is a violation of the postulated CP-symmetry: the combination of C-symmetry and P-symmetry . CP-symmetry states that the laws of physics should be the same if a particle were interchanged with its antiparticle , and left and right were swapped...
asymmetries in the case of Dirac neutrinos
Dirac fermion
In particle physics, a Dirac fermion is a fermion which is not its own anti-particle. It is named for Paul Dirac. All fermions in the standard model, except possibly neutrinos, are Dirac fermions...
(in the case of Majorana neutrinos
Majorana fermion
In physics, a Majorana fermion is a fermion which is its own anti-particle. The term is used in opposition to Dirac fermion, which describes particles that differ from their antiparticles...
, Majorana phases are still permitted, and could still lead to CP-violating effects).
For solar neutrinos
Solar neutrino problem
The solar neutrino problem was a major discrepancy between measurements of the numbers of neutrinos flowing through the Earth and theoretical models of the solar interior, lasting from the mid-1960s to about 2002...
the large angle MSW effect in tribimaximal mixing accounts for the experimental data
Solar neutrino problem
The solar neutrino problem was a major discrepancy between measurements of the numbers of neutrinos flowing through the Earth and theoretical models of the solar interior, lasting from the mid-1960s to about 2002...
, predicting average suppressions
in the Sudbury Neutrino ObservatorySudbury Neutrino Observatory
The Sudbury Neutrino Observatory is a neutrino observatory located 6,800 feet underground in Vale Inco's Creighton Mine in Sudbury, Ontario, Canada. The detector was designed to detect solar neutrinos through their interactions with a large tank of heavy water. The detector turned on in May 1999,...
(SNO) and
in lower energy solar neutrino experiments (and in long baseline reactor neutrino experiments). The bimaximally mixed in tribimaximal mixing accounts for the factor of two suppression observed for atmospheric muon-neutrinosNeutrino oscillation
Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvowhereby a neutrino created with a specific lepton flavor can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates...
(and confirmed in long-baseline accelerator experiments). Near-zero
appearance in a beam is predicted in exact tribimaximal mixing (), and future experimentsKamioka Observatory
The is a neutrino physics laboratory located underground in the Mozumi Mine of the Kamioka Mining and Smelting Co. near the Kamioka section of the city of Hida in Gifu Prefecture, Japan. A set of groundbreaking neutrino experiments have taken place at the observatory over the past two decades...
may well rule this out. Further characteristic predictions of tribimaximal mixing, e.g. for very long baseline
and (vacuum) survival probabilities , will be extremely hard to test experimentally.The L/E flatness of the electron-like event ratio at Super-Kamiokande
Super-Kamiokande
Super-Kamiokande is a neutrino observatory which is under Mount Kamioka near the city of Hida, Gifu Prefecture, Japan...
severely restricts the neutrino mixing matrices
to the form
Additional experimental data fixes
.The extension of this result to the CP violating case is found in reference .
History
The name tribimaximal first appeared in the literature in 2002 although this specific scheme had been previously published in 1999 as a viable alternative to the trimaximalTrimaximal mixing
Trimaximal mixing refers to the highly symmetric, maximally CP-violating, 3 \times 3 fermion mixing configuration, characterised by a unitary matrix having all its elements equal in modulus...
scheme. Tribimaximal mixing is sometimes confused with other mixing schemes, e.g. which differ from tribimaximal mixing by row- and/or column-wise permutations of the mixing-matrix elements. Such permuted forms are experimentally distinct however, and are now ruled out by data.
That the L/E flatness of the electron-like event ratio at Superkamiokande severely restricts the neutrino mixing matrices was first presented by D. V. Ahluwalia in a Nuclear and Particle Physics Seminar of the Los Alamos National Laboratory
Los Alamos National Laboratory
Los Alamos National Laboratory is a United States Department of Energy national laboratory, managed and operated by Los Alamos National Security , located in Los Alamos, New Mexico...
on June 5, 1998. It was just a few hours after the Super-Kamiokande
Super-Kamiokande
Super-Kamiokande is a neutrino observatory which is under Mount Kamioka near the city of Hida, Gifu Prefecture, Japan...
press conference that announced the results on atmospheric neutrinos.