Solution to the strong CP problem with gauge-mediated supersymmetry breaking

We demonstrate that a certain class of low scale supersymmetric ``Nelson-Barr'' type models can solve the strong and supersymmetric CP problems while at the same time generating sufficient weak CP violation in the $K^{0}-\bar{K}^{0}$ system. In order to prevent one-loop corrections to $\bar{\theta}$ which violate bounds coming from the neutron electric dipole moment (EDM), one needs a scheme for the soft supersymmetry breaking parameters which can naturally give sufficient squark degeneracies and proportionality of trilinear soft supersymmetry-breaking parameters to Yukawa couplings. We show that a gauge-mediated supersymmetry breaking sector can provide the needed degeneracy and proportionality, though that proves to be a problem for generic Nelson-Barr models. The workable model we consider here has the Nelson-Barr mass texture enforced by a gauge symmetry; one also expects a new U(1) gauge superfield with mass in the TeV range. The resulting model is predictive. We predict a measureable neutron EDM and the existence of extra vector-like quark superfields which can be discovered at the LHC. Because the $3\times 3$ Cabbibo-Kobayashi-Maskawa matrix is approximately real, the model also predicts a flat unitarity triangle and the absence of substantial CP violation in the $B$ system at future $B$ factories. We discuss the general issues pertaining to the construction of such a workable model and how they lead to the successful strategy. A detailed renormalization group study is then used to establish the feasibility of the model considered