Quark masses and mixing in the D_5 model under spontaneous CP violation

It is known that CP violation occurs in flavor physics, and the CKM matrix is complex. We investigate the Yukawa sector of a four-Higgs model based on D_5 symmetry. To understand all possible sources of CP violation in the Yukawa sector, we systematically analyze the impact of all possible representation assignments of the left-handed and right-handed quark fields under the D_5 group. After imposing the conditions of non-block-diagonal CKM matrix and the absence of massless quarks, we find that the only viable representation consists of one D_5 doublet and one singlet. In this case, two distinct types of quark mass models emerge in the quark sector. For each model, we introduce the most general vacuum that allows spontaneous CP violation (SCPV) into the mass matrices. Due to the specific structure of the D_5 symmetry, the vacuum expectation values (vevs) satisfy particular relations such that, for both quark models, the up- and down-quark mass squared matrices (i.e., M_d M_d^\dagger and M_u M_u^\dagger) can be simultaneously block-diagonalized by the same unitary transformation, forcing the CKM matrix to be block-diagonal and thus yielding vanishing mixing angles and CP violation. This result rules out the possibility of explaining experimental observations within the D_5 4HDM under the SCPV framework.