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We introduce a class of multi-Higgs doublet extensions of the Standard Model that solve the strong $CP$problem with profound consequences for the flavor sector. The Yukawa matrices are constrained to have many zero entries by a “Higgs-flavor” symmetry, $G_{\mathrm{HF}}$, that acts on Higgs and quark fields. The violation of both $CP$and $G_{\mathrm{HF}}$occurs in the Higgs mass matrix so that, for certain choices of $G_{\mathrm{HF}}$charges, the strong $CP$parameter $\overline{\theta }$is zero at tree level. Radiative corrections to $\overline{\theta }$are computed in this class of theories. They vanish in realistic two-Higgs doublet models with $G_{\mathrm{HF}}={\mathbb{Z}}_3$. We also construct realistic three-Higgs models with $G_{\mathrm{HF}}=\mathrm{U}\left(1\right)$, where the one-loop results for $\overline{\theta }$are model-dependent. Requiring $\overline{\theta }<{10}^{-10}$has important implications for the flavor problem by constraining the Yukawa coupling and Higgs mass matrices. Contributions to $\overline{\theta }$from higher-dimension operators are computed at one loop and can also be sufficiently small, although the hierarchy problem of this class of theories is worse than in the Standard Model. 

Abstract The field of particle physics is at the crossroads. The discovery of a Higgs-like boson completed the Standard Model (SM), but the lacking observation of convincing resonances Beyond the SM (BSM) offers no guidance for the future of particle physics. On the other hand, the motivation for New Physics has not diminished and is, in fact, reinforced by several striking anomalous results in many experiments. Here we summarise the status of the most significant anomalies, including the most recent results for the flavour anomalies, the multi-lepton anomalies at the LHC, the Higgs-like excess at around 96 GeV, and anomalies in neutrino physics, astrophysics, cosmology, and cosmic rays. While the LHC promises up to 4 $$\hbox {ab}^{-1}$$ ab - 1 of integrated luminosity and far-reaching physics programmes to unveil BSM physics, we consider the possibility that the latter could be tested with present data, but that systemic shortcomings of the experiments and their search strategies may preclude their discovery for several reasons, including: final states consisting in soft particles only, associated production processes, QCD-like final states, close-by SM resonances, and SUSY scenarios where no missing energy is produced. New search strategies could help to unveil the hidden BSM signatures, devised by making use of the CERN open data as a new testing ground. We discuss the CERN open data with its policies, challenges, and potential usefulness for the community. We showcase the example of the CMS collaboration, which is the only collaboration regularly releasing some of its data. We find it important to stress that individuals using public data for their own research does not imply competition with experimental efforts, but rather provides unique opportunities to give guidance for further BSM searches by the collaborations. Wide access to open data is paramount to fully exploit the LHCs potential.