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The exploration of the phase diagram of Quantum ChromoDynamics (QCD) iscarried out by studying ultrarelativistic heavy-ion collisions. The energyrange covered by the CERN SPS ($\sqrt{s_{\rm \scriptscriptstyle{NN}}} \sim$6-17 GeV) is ideal for the investigation of the region of the phase diagramcorresponding to finite baryochemical potential ($\mu_{\rm B}$), and has beenlittle explored up to now. We propose in this document a new experiment, NA60+,that would address several observables which are fundamental for theunderstanding of the phase transition from hadronic matter towards aQuark-Gluon Plasma (QGP) at SPS energies. In particular, we propose to study,as a function of the collision energy, the production of thermal dimuons fromthe created system, from which one would obtain a caloric curve of the QCDphase diagram that is sensitive to the order of the phase transition. Inaddition, the measurement of a $\rho$-a$_1$ mixing contribution would provideconclusive insights into the restoration of the chiral symmetry of QCD. Inparallel, studies of heavy quark and quarkonium production would also becarried out, addressing the measurement of transport properties of the QGP andthe investigation of the onset of the deconfinement transition. The documentalso defines an experimental set-up which couples a vertex telescope based onmonolithic active pixel sensors (MAPS) to a muon spectrometer with tracking(GEM) and triggering (RPC) detectors within a large acceptance toroidal magnet.Results of physics performance studies for most observables accessible to NA60+are discussed, showing that the results of the experiment would lead to asignificant advance of our understanding of strong interaction physics. Thedocument has been submitted as an input to the European Particle PhysicsStrategy Update 2018-2020 (http://europeanstrategyupdate.web.cern.ch/).

Study of hard and electromagnetic processes at CERN-SPS energies: an  investigation of the high-$\mu_{\mathbf{B}}$ region of the QCD phase diagram  with NA60+