Viability of minimal left-right models with discrete symmetries

We provide a systematic study of minimal left-right models that are invariant under $P$, $C$, and/or $CP$ transformations. Due to the high amount of symmetry such models are quite predictive in the amount and pattern of $CP$ violation they can produce or accommodate at lower energies. Using current experimental constraints some of the models can already be excluded. For this purpose we provide an overview of the experimental constraints on the different left-right symmetric models, considering bounds from colliders, meson-mixing and low-energy observables, such as beta decay and electric dipole moments. The features of the various Yukawa and Higgs sectors are discussed in detail. In particular, we give the Higgs potentials for each case, discuss the possible vacua and investigate the amount of fine-tuning present in these potentials. It turns out that all left-right models with $P$, $C$, and/or $CP$ symmetry have a high degree of fine-tuning, unless supplemented with mechanisms to suppress certain parameters. The models that are symmetric under both $P$ and $C$ are not in accordance with present observations, whereas the models with either $P$, $C$, or $CP$ symmetry can not be excluded by data yet. To further constrain and discriminate between the models measurements of $B$-meson observables at LHCb and $B$-factories will be especially important, while measurements of the EDMs of light nuclei in particular could provide complementary tests of the LRMs.