# Quark–lepton complementarity

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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).

## Possible evidence for QLC

Recent[ when? ] neutrino experiments confirm that the Pontecorvo–Maki–Nakagawa–Sakata matrix UPMNS contains large[ clarification needed ] mixing angles. For example, atmospheric measurements of particle decay yield
θPMNS
23
45°, while solar experiments yield
θPMNS
12
34°. These results should be compared with
θPMNS
13
which is small, [2] and with the quark mixing angles in the Cabibbo–Kobayashi–Maskawa matrix UCKM. 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.

${\displaystyle \theta _{12}^{PMNS}+\theta _{12}^{CKM}\approx 45^{\circ }\ ,}$
${\displaystyle \quad \quad \theta _{23}^{PMNS}+\theta _{23}^{CKM}\approx 45^{\circ }\ .}$

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 VM is simply defined as the product of the CKM and PMNS matrices:

${\displaystyle V_{\mathrm {M} }=U_{\mathrm {CKM} }\cdot U_{\mathrm {PMNS} }\ ,}$

Unitarity implies:

${\displaystyle U_{\mathrm {PMNS} }=U_{\mathrm {CKM} }^{\dagger }V_{\mathrm {M} }\ .}$

## Open questions

One may ask where do the large lepton mixings come from? Is this information implicit in the form of the VM 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 ${\displaystyle V_{M}}$ 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 VM in models with renormalization effects that are relevant, however, only in particular cases with tanβ > 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:

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,
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, and
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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.

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W0
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η
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## References

1. 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.
2. 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

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