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.
Categories: hep-ph