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:
:
This mixing is historically interesting as it is quite close to reality when compared to other simple hypotheses where the squares of matrix elements take exact ratios, and also compared to the naive supposition that the matrix would be approximately diagonal like the
CKM matrix. However the precision of modern experiments mean that such a simple form is
excluded by experiment at a level of over 5σ, mainly due to the fact the tribimaximal scheme has a zero in the
element, but also (to a much lesser extent) because it predicts no violation of
CP symmetry
In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry (parity symmetry). CP-symmetry states that the laws of physics should be the ...
.
The tribimaximal mixing form was compatible with pre-2011
neutrino oscillation experiments
[
] and may be used as a
zeroth-order approximation
In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is.
Usage in science and engineering
In formal expressions, the ordinal number used ...
to more general forms for the PMNS matrix,
[
][
] including some that are consistent with the data. In the PDG
convention for the PMNS matrix, tribimaximal mixing may be specified in terms of
lepton mixing angles as follows:
:
The above prediction has been falsified experimentally, because ''θ''
13 was found to be nontrivial, ''θ''
13 =8.5°.
A non-negligible value of ''θ''
13 has been foreseen in certain theoretical schemes that were put forward before tribimaximal mixing and that
supported a large solar mixing, before it was confirmed experimentally
[
] (these theoretical schemes do not have a special name, but for the reasons explained above, they could be called pre-tribimaximal or also non-tribimaximal). This situation is not new: also in the 1990s, the solar mixing angle was supposed to be small by most theorists, until KamLAND proved the contrary to be true.
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
trimaximal[
] and
bimaximal[
][
] mixing schemes, both now ruled out by data. In tribimaximal mixing,
the
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
mass
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
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.
[
]
Phenomenology
By virtue of the zero (
) 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
A solar neutrino is a neutrino originating from nuclear fusion in the Sun's core, and is the most common type of neutrino passing through any source observed on Earth at any particular moment. Neutrinos are elementary particles with extremely smal ...
the large angle
MSW effect
MSW may stand for: Businesses and organisations
* Manatawny Still Works, a craft distillery in Pottstown, Pennsylvania, United States
* Massawa International Airport, Eritrea (by IATA code)
* Ministry of Interior (Poland) ( pl, Ministerstwo Spraw ...
in tribimaximal mixing accounts for the
experimental data
Experimental data in science and engineering is data produced by a measurement, test method, experimental design or quasi-experimental design. In clinical research any data produced are the result of a clinical trial. Experimental data may be qua ...
, predicting average suppressions
in the
Sudbury Neutrino Observatory
The Sudbury Neutrino Observatory (SNO) was a neutrino observatory located 2100 m underground in Vale's Creighton Mine in Sudbury, Ontario, Canada. The detector was designed to detect solar neutrinos through their interactions with a large t ...
(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 this has been strongly ruled out by modern
reactor neutrino experiments. 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 (abbreviation of Super-Kamioka Neutrino Detection Experiment, also abbreviated to Super-K or SK; ja, スーパーカミオカンデ) is a neutrino observatory located under Mount Ikeno near the city of Hida, Gifu Prefecture ...
severely restricts the neutrino mixing matrices to the form given by Stancu & Ahluwalia (1999):[
]
:
Additional experimental data fixes The extension of this result to the CP-violating case is found in Ahluwalia, Liu, & Stancu (2002).[
]
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 trimaximal 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 (often shortened as Los Alamos and LANL) is one of the sixteen research and development laboratories of the United States Department of Energy (DOE), located a short distance northwest of Santa Fe, New Mexico, ...
on June 5, 1998. It was just a few hours after the Super-Kamiokande
Super-Kamiokande (abbreviation of Super-Kamioka Neutrino Detection Experiment, also abbreviated to Super-K or SK; ja, スーパーカミオカンデ) is a neutrino observatory located under Mount Ikeno near the city of Hida, Gifu Prefecture ...
press conference that announced the results on atmospheric neutrinos.
References
Leptons
Standard Model
Particle physics
Neutrinos
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