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Compositeness provides an elegant fundamental mechanism for electroweak symmetry breaking. The essential ingredient is the existence of a new strong force that breaks chiral symmetry. Through the gauging of an appropriate SU(2) x U(1) subgroup of the residual symmetry group, the Nambu-Goldstone boson associated to the symmetry breaking becomes the Standard Model Higgs. The Sp(4) gauge theory with two Dirac fundamental flavours provides a candidate for the microscopic origin of composite Higgs models based on the SU(4)/Sp(4) coset of the chiral symmetry breaking, which has been advocated as having a phenomenology compatible with the Standard Model. We report on the first non-perturbative calculation from first principles of this theory. We present the first results for the spectrum of glueballs of the pure Sp(4) Yang-Mills theory, an EFT framework for the interpretation of the masses and decay constants of the lightest pion, vector and axial-vector mesons, and a preliminary calculation of the latter in the quenched approximation. We conclude by introducing the technical steps necessary to set up future investigations of the mesonic spectrum of the full theory.