SpontaneousCPviolation inE6supersymmetric grand unified theory with SU(2) flavor and anomalous U(1) symmetries

Abstract

We construct a model of spontaneous $CP$ violation in ${\mathrm{E}}_{6}$ supersymmetric grand unified theory. In the model, we employ an $\mathrm{SU}(2{)}_{\mathrm{F}}$ flavor symmetry and an anomalous $\mathrm{U}(1{)}_{\mathrm{A}}$ symmetry. The $\mathrm{SU}(2{)}_{\mathrm{F}}$ flavor symmetry is introduced to provide the origin of hierarchical structures of Yukawa coupling and to ensure the universality of sfermion soft masses. The anomalous $\mathrm{U}(1{)}_{\mathrm{A}}$ symmetry is introduced to realize the doublet-triplet mass splitting, to provide the origin of hierarchical structures of Yukawa couplings, and to solve the $\ensuremath{\mu}$ problem. In the model, $CP$ is spontaneously broken by the $\mathrm{SU}(2{)}_{\mathrm{F}}$ breaking in order to provide a Kobayashi-Maskawa phase and to evade the supersymmetric $CP$ problem. However, a naive construction of the model generally leads to an unwanted outcome, $\mathrm{arg}[\ensuremath{\mu}{b}^{*}]=\mathcal{O}(1)$, when $CP$ violating effects in the flavor sector are taken into account. We cure this difficulty by imposing a discrete symmetry and find that this prescription can play additional roles. It ensures that the realistic up-quark mass and Cabibbo angle are simultaneously realized without cancellation between $\mathcal{O}(1)$ coefficients. Also, severe constraints from the chromo-electric dipole moment of the quark can be satisfied without destabilizing the weak scale. The discrete symmetry reduces the number of free parameters, but the model is capable of reproducing quark and lepton mass spectra, mixing angles, and a Jarlskog invariant. We obtain characteristic predictions ${V}_{ub}\ensuremath{\sim}\mathcal{O}({\ensuremath{\lambda}}^{4})$ ($\ensuremath{\lambda}=0.22$) and $|{V}_{cb}{Y}_{b}|=|{Y}_{c}|$ at the grand unified theory scale.