A dynamical CP source for CKM, PMNS and Leptogenesis

We propose a simple model for the spontaneous CP violation based on $SU(2)_{L}\times U(1)_{Y}\times A_{4}\times CP\times Z_{2}$ symmetry for quarks and leptons in a seesaw framework. In our model CP is spontaneously broken at high energies, after breaking of flavor symmetry, by a complex vacuum expectation value of $A_{4}$-triplet and gauge singlet scalar field. And, certain effective dimension-5 operators are introduced in the quark Lagrangian as an equal footing, which lead the quark mixing matrix to the Cabibbo-Kobayashi-Maskawa (CKM) one in the form, while the lepton Lagrangian is renormalizable, which gives rise to a non-degenerate Dirac neutrino Yukawa matrix and an unique CP-phase, and which in turn explains the non-zero value of $\theta_{13}\simeq9^{\circ}$ and two large mixing angles $\theta_{12}, \theta_{23}$. We show that the spontaneously generated CP phase "$\xi$" could become a natural source of leptogenesis as well as CP violations in the CKM and Pontecorvo-Maki-Nakagawa-Sakata (PMNS). Moreover, we provide a solution for the strong CP problem. Interestingly enough, we show that, for around $\xi\simeq120^{\circ}$, the quarks lead to the correct CKM CP-phase corresponding to $\delta^{q}_{CP}\simeq67^{\circ}$, while the leptons with the measured value of $\theta_{13}$ favor $\delta_{CP}=$, $|\theta_{23}-45^{\circ}|\rightarrow0$ for normal mass hierarchy and $\delta_{CP}=$, $|\theta_{23}-45^{\circ}|\rightarrow5^{\circ}$ for inverted one. As a numerical study in the lepton sector, we show low-energy phenomenologies and leptogenesis for the normal and inverted case, respectively, and a link between them.