Nontrivial Correlation between the CKM and MNS Matrices

We point out that the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix $V_{\rm CKM}$ and the Maki-Nakagawa-Sakata (MNS) lepton mixing matrix $V_{\rm MNS}$ can naturally be correlated in a class of seesaw models with grand unification, but the texture of their correlation matrix ${\cal F}_\nu$ is rather nontrivial. The bimaximal mixing pattern of ${\cal F}_\nu$ is disfavored by current data, and other special forms of ${\cal F}_\nu$ may suffer from fine-tuning of the free phase parameters in fitting the so-called quark-lepton complementarity relation. A straightforward calculation of ${\cal F}_\nu$ in terms of $V_{\rm CKM}$ and $V_{\rm MNS}$ reveals a striking feature of ${\cal F}_\nu$: its (1,3) element cannot be zero or too small, no matter whether the (1,3) elements of $V_{\rm CKM}$ and $V_{\rm MNS}$ are vanishing or not. We also add some brief comments on possible radiative corrections to $V_{\rm CKM}$ and $V_{\rm MNS}$.