Pati-Salam unification with a spontaneous $CP$ violation

Recent neutrino oscillation experiments suggest that the Pontecorvo-Maki-Nakagawa-Sakata matrix in the lepton sector has a $CP$ violating phase like the Cabibbo-Kobayashi-Maskawa matrix in the quark sector. However, the origins of these phases in both matrices are not clarified by now. Although complex Yukawa couplings could induce these phases, the phases remain as free parameters of the model even in that case. If the $CP$ symmetry is considered to be spontaneously broken, they are expected to be determined by some physics at a much lower energy scale than the Planck scale. We study such a possibility in a framework of Pati-Salam-type unification. We also discuss other phenomenological issues in it.

Figure 1

Left: A diagram for the neutrino mass generation due to the type I seesaw in the minimal model. Right: A one-loop diagram for the neutrino mass generation due to the scotogenic type III seesaw in the extended model.

Figure 2

Left panel: A typical solution of the Boltzmann equations for $Y_{N_1}$ and $Y_{N_3}$. Their equilibrium values $Y_{N_1}^{\mathrm{eq}}$ and $Y_{N_3}^{\mathrm{eq}}$ are also plotted in the same panel. While $Y_{N_3}$ is found to follow $Y_{N_3}^{\mathrm{eq}}$ at $z\gtrsim 0.4$, $Y_{N_1}$ keeps a constant value until $N_1$ starts decaying. Right panel: The evolution of the lepton number asymmetry $Y_L$ generated through the out-of-equilibrium decay of $N_1$. Horizontal dotted lines show the value of $Y_L$ required in this model to realize the baryon number asymmetry in the Universe [20].

Figure 3

The $N_1$ decay diagrams which contribute to the generation of the lepton number asymmetry. The interference between them causes the $CP$ asymmetry $ϵ$.