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This paper describes the physics case for a new fixed target facility at CERN SPS. The SHiP (search for hidden particles) experiment is intended to hunt for new physics in the largely unexplored domain of very weakly interacting particles with masses below the Fermi scale, inaccessible to the LHC experiments, and to study tau neutrino physics. The same proton beam setup can be used later to look for decays of tau-leptons with lepton flavour number non-conservation, [Formula: see text] and to search for weakly-interacting sub-GeV dark matter candidates. We discuss the evidence for physics beyond the standard model and describe interactions between new particles and four different portals-scalars, vectors, fermions or axion-like particles. We discuss motivations for different models, manifesting themselves via these interactions, and how they can be probed with the SHiP experiment and present several case studies. The prospects to search for relatively light SUSY and composite particles at SHiP are also discussed. We demonstrate that the SHiP experiment has a unique potential to discover new physics and can directly probe a number of solutions of beyond the standard model puzzles, such as neutrino masses, baryon asymmetry of the Universe, dark matter, and inflation.

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The standard picture of viable higher-dimensional theories is that direct manifestations of extra dimensions occur at short distances only, whereas long-distance physics is described by effective four-dimensional theories. We show that this is not necessarily true in models with infinite extra dimensions.As an example, we consider a five-dimensional scenario with three 3-branes in which gravity is five-dimensional both at short and very long distance scales, with conventional four-dimensional gravity operating at intermediate length scales. A phenomenologically acceptable range of validity of four-dimensional gravity extending from microscopic to cosmological scales is obtained without strong fine-tuning of parameters.