Spontaneous CP violation in a SUSY model with a complex CKM

It is pointed out that the recent measurement of the angle γ of the unitarity triangle, providing irrefutable evidence for a complex Cabibbo-Kobayashi-Maskawa (CKM) matrix, presents a great challenge for supersymmetric models with spontaneous CP violation. We construct a new minimal extension of the minimal supersymmetric standard model (MSSM), with spontaneous CP breaking, which leads to a complex CKM matrix, thus conforming to present experimental data. This is achieved through the introduction of two singlet chiral superfields and a vector-like quark chiral superfield which mixes with the standard quarks. A Z3 symmetry is introduced in order to have a potential solution to the strong CP problem.