How natural is a smallθ¯in left-right supersymmetry models?

In a world without an axion, the smallness of $\overline{\theta }$ may be achieved due to a spontaneously broken discrete left-right symmetry. We analyze the radiatively induced $\overline{\theta }$ in the context of generic left-right symmetric supersymmetry models without assuming flavor degeneracy in the squark sector. Left-right symmetry allows us to keep $\overline{\theta }$ within its present bound only if the intergenerational mass splitting in the squark sector at the scale of the left-right symmetry breaking is smaller than 0.5%. We also consider the naturalness of the $m_u=0\mathrm{}$ solution to the strong CP problem in the context of horizontal flavor symmetries. A strong bound on the combination of the horizontal charges in the up quark sector is found in this case.