Supersymmetry and the Nelson-Barr mechanism

One possible solution to the strong $\mathrm{CP}$ problem is that $\mathrm{CP}$ is an exact symmetry, spontaneously broken at some scale. Some years ago, Nelson and Barr suggested a mechanism for obtaining $\theta =0\mathrm{}$ at the tree level in this framework and showed that radiative corrections were small in some nonsupersymmetric models. Further investigations suggested that the same could be true in supersymmetric theories. In this paper we show that such solutions assume extraordinarily high degrees of degeneracy among squark masses and among other supersymmetry-breaking parameters. We argue, using naturalness as well as expectations from string theory, that this is not very plausible.