Quark mixing with soft breaking of the parity in the minimal Left-Right model

We study the possibility of soft breaking effects of the generalized parity within the minimal Left-Right model. One aim of the paper is to elaborate on the potentiality, the limit, and the predictivity of a restored parity at high scale. While revisiting the issue of strong CP in the Left-Right theories, we motivate the possibility of explicit-parity-breaking, that we then parameterize in the right-handed quark mixing matrix. The strong CP parameter $\bar{\theta}$ is also parameterized in terms of the breaking. We discuss some possible phenomenological consequences in this scenario. In particular, the constraint provided by $\bar{\theta}$ enables us to quantify the maximal deviation of the right-handed quark mixings from the standard case with exact parity in term of a single parameter. This deviation has a direct impact on flavor physics.