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The **quark–lepton complementarity** (**QLC**) is a possible fundamental symmetry between quarks and leptons. First proposed in 1990 by Foot and Lew,^{ [1] } it assumes that leptons as well as quarks come in three "colors". Such theory may reproduce the Standard Model at low energies, and hence quark–lepton symmetry may be realized in nature.

A **quark** is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as *color confinement*, quarks are never directly observed or found in isolation; they can be found only within hadrons, which include baryons and mesons. For this reason, much of what is known about quarks has been drawn from observations of hadrons.

In particle physics, a **lepton** is an elementary particle of half-integer spin that does not undergo strong interactions. Two main classes of leptons exist: charged leptons, and neutral leptons. Charged leptons can combine with other particles to form various composite particles such as atoms and positronium, while neutrinos rarely interact with anything, and are consequently rarely observed. The best known of all leptons is the electron.

**Color charge** is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD).

Recent^{[ when? ]} neutrino experiments confirm that the Pontecorvo–Maki–Nakagawa–Sakata matrix *U*_{PMNS} contains large^{[ clarification needed ]} mixing angles. For example, atmospheric measurements of particle decay yield ^{}*θ*^{PMNS}_{23} ≈ 45°, while solar experiments yield ^{}*θ*^{PMNS}_{12} ≈ 34°. These results should be compared with ^{}*θ*^{PMNS}_{13} which is small,^{ [2] } and with the quark mixing angles in the Cabibbo–Kobayashi–Maskawa matrix *U*_{CKM}. The disparity that nature indicates between quark and lepton mixing angles has been viewed in terms of a "quark–lepton complementarity" which can be expressed in the relations

A **neutrino** is a fermion that interacts only via the weak subatomic force and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small (*-ino*) that it was long thought to be zero. The mass of the neutrino is much smaller than that of the other known elementary particles. The weak force has a very short range, the gravitational interaction is extremely weak, and neutrinos, as leptons, do not participate in the strong interaction. Thus, neutrinos typically pass through normal matter unimpeded and undetected.

In particle physics, the **Pontecorvo–Maki–Nakagawa–Sakata matrix**, **Maki–Nakagawa–Sakata matrix**, **lepton mixing matrix**, or **neutrino mixing matrix** is a unitary mixing matrix which contains information on the mismatch of quantum states of neutrinos when they propagate freely and when they take part in the weak interactions. It is a model of neutrino oscillation. This matrix was introduced in 1962 by Ziro Maki, Masami Nakagawa and Shoichi Sakata, to explain the neutrino oscillations predicted by Bruno Pontecorvo.

Possible consequences of QLC have been investigated in the literature and in particular a simple correspondence between the PMNS and CKM matrices have been proposed and analyzed in terms of a correlation matrix. The correlation matrix *V*_{M} is simply defined as the product of the CKM and PMNS matrices:

Unitarity implies:

One may ask where do the large lepton mixings come from? Is this information implicit in the form of the *V*_{M} matrix? This question has been widely investigated in the literature, but its answer is still open. Furthermore, in some Grand Unification Theories (GUTs) the direct QLC correlation between the CKM and the PMNS mixing matrix can be obtained. In this class of models, the matrix is determined by the heavy Majorana neutrino mass matrix.

Despite the naive relations between the PMNS and CKM angles, a detailed analysis shows that the correlation matrix is phenomenologically compatible with a tribimaximal pattern, and only marginally with a bimaximal pattern. It is possible to include bimaximal forms of the correlation matrix *V*_{M} in models with renormalization effects that are relevant, however, only in particular cases with tan*β* > 40 and with quasi-degenerate neutrino masses.

**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:

In particle physics, the **electroweak interaction** is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different aspects of the same force. Above the unification energy, on the order of 246 GeV, they would merge into a single **electroweak force**. Thus, if the universe is hot enough (approximately 10^{15} K, a temperature not exceeded since shortly after the Big Bang), then the electromagnetic force and weak force merge into a combined electroweak force. During the quark epoch, the electroweak force split into the electromagnetic and weak force.

The **W and Z bosons** are together known as the **weak** or more generally as the **intermediate vector bosons.** These elementary particles mediate the weak interaction; the respective symbols are ^{}W^{+}_{}, ^{}W^{−}_{}, and ^{}Z^{}. The W bosons have either a positive or negative electric charge of 1 elementary charge and are each other's antiparticles. The Z boson is electrically neutral and is its own antiparticle. The three particles have a spin of 1. The W bosons have a magnetic moment, but the Z has none. All three of these particles are very short-lived, with a half-life of about 3×10^{−25} s. Their experimental discovery was a triumph for what is now known as the Standard Model of particle physics.

**R-parity** is a concept in particle physics. In the Minimal Supersymmetric Standard Model, baryon number and lepton number are no longer conserved by all of the renormalizable couplings in the theory. Since baryon number and lepton number conservation have been tested very precisely, these couplings need to be very small in order not to be in conflict with experimental data. R-parity is a symmetry acting on the Minimal Supersymmetric Standard Model (MSSM) fields that forbids these couplings and can be defined as

In the Standard Model of particle physics, the **Cabibbo–Kobayashi–Maskawa matrix**, **CKM matrix**, **quark mixing matrix**, or **KM matrix** is a unitary matrix which contains information on the strength of the flavour-changing weak interaction. Technically, it specifies the mismatch of quantum states of quarks when they propagate freely and when they take part in the weak interactions. It is important in the understanding of CP violation. This matrix was introduced for three generations of quarks by Makoto Kobayashi and Toshihide Maskawa, adding one generation to the matrix previously introduced by Nicola Cabibbo. This matrix is also an extension of the GIM mechanism, which only includes two of the three current families of quarks.

A **chiral** phenomenon is one that is not identical to its mirror image. The spin of a particle may be used to define a **handedness**, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity transformation. Invariance under parity transformation by a Dirac fermion is called **chiral symmetry**.

**Neutrino oscillation** is a quantum mechanical phenomenon whereby a neutrino created with a specific lepton family number can later be measured to have a different lepton family number. The probability of measuring a particular flavor for a neutrino varies between 3 known states, as it propagates through space.

In particle physics, **flavour** or **flavor** refers to the *species* of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with *flavour quantum numbers* that are assigned to all subatomic particles. They can also be described by some of the family symmetries proposed for the quark-lepton generations.

The **Weinberg angle** or **weak mixing angle** is a parameter in the Weinberg–Salam theory of the electroweak interaction, part of the Standard Model of particle physics, and is usually denoted as *θ*_{W}. It is the angle by which spontaneous symmetry breaking rotates the original ^{}W^{0}_{} and B^{0} vector boson plane, producing as a result the ^{}Z^{0}_{} boson, and the photon.

This article describes the mathematics of the **Standard Model** of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as containing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs particle.

**Sterile neutrinos** are hypothetical particles that interact only via gravity and do not interact via any of the fundamental interactions of the Standard Model. The term *sterile neutrino* is used to distinguish them from the known *active neutrinos* in the Standard Model, which are charged under the weak interaction.

The **Daya Bay Reactor Neutrino Experiment** is a China-based multinational particle physics project studying neutrinos. The multinational collaboration includes researchers from China, Chile, the United States, Taiwan, Russia, and the Czech Republic. The US side of the project is funded by the US Department of Energy's Office of High Energy Physics.

**Chooz** (French: [ʃo]) was a short baseline neutrino oscillation experiment in Chooz, France. Its major result was setting limits on the neutrino oscillation parameters responsible for changing electron neutrinos into other neutrinos. Specifically, it found that sin^{2}(2*θ*_{13}) < 0.17 for large δm^{2} and δm^{2} > 8×10^{−4} eV^{2} for maximal mixing. Results were published in 1999.

The **Koide formula** is an unexplained empirical equation discovered by Yoshio Koide in 1981. In its original form, it relates the masses of the three charged leptons; later authors have extended the relation to neutrinos, quarks, and other families of particles.

**Trimaximal mixing** refers to the highly symmetric, maximally CP-violating, fermion mixing configuration, characterised by a unitary matrix having all its elements equal in modulus ( , ) as may be written, e.g.:

The **eta** and **eta prime meson** are isosinglet mesons made of a mixture of up, down and strange quarks and their antiquarks. The charmed eta meson and bottom eta meson are similar forms of quarkonium; they have the same spin and parity as the (light) ^{}η^{} defined, but are made of charm quarks and bottom quarks respectively. The top quark is too heavy to form a similar meson, due to its very fast decay.

In particle physics, **CP violation** is a violation of **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 is interchanged with its antiparticle while its spatial coordinates are inverted. The discovery of CP violation in 1964 in the decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch.

**Double Chooz** is a short-baseline neutrino oscillation experiment in Chooz, France. Its goal is to measure or set a limit on the *θ*_{13} mixing angle, a neutrino oscillation parameter responsible for changing electron neutrinos into other neutrinos. The experiment uses reactors of the Chooz Nuclear Power Plant as a neutrino source and measures the flux of neutrinos they receive. To accomplish this, Double Chooz has a set of two detectors situated 400 meters and 1050 meters from the reactors. Double Chooz is a successor to the Chooz experiment; one of its detectors occupies the same site as its predecessor. Until January 2015 all data has been collected using only the far detector. The near detector was completed in September 2014, after construction delays and is taking physics data since beginning of 2015.

- ↑ R. Foot, H. Lew (1990). "Quark-lepton-symmetric model".
*Physical Review D*.**41**(11): 3502–3505. Bibcode:1990PhRvD..41.3502F. doi:10.1103/PhysRevD.41.3502. - ↑ F. P. An et al. [DAYA-BAY Collaboration], Phys. Rev. Lett. 108, 171803 (2012) [arXiv:1203.1669 [hep-ex]] https://arxiv.org/abs/arXiv:1203.1669

- B.C. Chauhan; M. Picariello; J. Pulido; E. Torrente-Lujan (2007). "Quark-lepton complementarity, neutrino and standard model data predict
*θ*^{PMNS}_{13}= (9+1

−2)°".*European Physical Journal C*.**50**(3): 573–578. arXiv: hep-ph/0605032 . Bibcode:2007EPJC...50..573C. doi:10.1140/epjc/s10052-007-0212-z.

The * European Physical Journal C* (

**arXiv** is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, physics, astronomy, electrical engineering, computer science, quantitative biology, statistics, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, and had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month.

The **bibcode** is a compact identifier used by several astronomical data systems to uniquely specify literature references.

- K.M. Patel (2011). "An
*SO*(10) ×*S*_{4}Model of Quark-Lepton Complementarity;".*Physics Letters B*.**695**(1–4): 225–230. arXiv: 1008.5061 . Bibcode:2011PhLB..695..225P. doi:10.1016/j.physletb.2010.11.024.

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