Quark-Lepton complementarity

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Recent neutrino experiments confirm that the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton mixing matrix $U P M N S$ contains large mixing angles. For example the atmospheric mixing $\theta_{23}^{PMNS}$ is compatible with $45^\circ$ , and the solar mixing $\theta_{12}^{PMNS}$ is $\approx 34^\circ$ . These results should be compared with the third lepton mixing angle $\theta_{13}^{PMNS}$ which is very small and even compatible with zero, and with the quark mixing angles in the Cabibbo-Kobayashi-Maskawa matrix $U C K M$ .

The disparity that nature indicates between quark and lepton mixing angles has been viewed in terms of a 'quark-lepton complementarity' (QLC) which can be expressed in the relations

$\theta_{12}^{PMNS}+\theta_{12}^{CKM}\simeq 45^\circ\,; \quad\quad \theta_{23}^{PMNS}+\theta_{23}^{CKM}\simeq 45^\circ\,.$

Possible consequences of QLC have been investigated in the literature and in particular a simple correspondence between the $U P M N S$ and $U C K M$ matrices has 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, $V_M=U_{CKM}\cdot U_{PMNS}$ , and efforts have been done to obtain the most favorite pattern for the matrix $V M$ . Unitarity then implies $U_{PMNS}=U^{\dagger}_{CKM}V_M$ and 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 $V M$ 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 $V M$ in models with renormalization effects that are relevant, however, only in particular cases with large $tanβ( > 40)$ and with quasi degenerate neutrino masses.