$Sp\left(4\right)$ gauge theory on the lattice: towards $SU\left(4\right)/Sp\left(4\right)$ composite Higgs (and beyond)

The $Sp\left(4\right)$ gauge theory with two Dirac fundamental flavours provides a candidate for the microscopic origin of composite-Higgs models based on the $SU\left(4\right)/Sp\left(4\right)$ coset. We employ a combination of two different, complementary strategies for the numerical lattice calculations, based on the Hybrid Monte Carlo and on the Heat Bath algorithms. We perform pure Yang-Mills, quenched computations and exploratory studies with dynamical Wilson fermions. We present the first results in the literature for the spectrum of glueballs of the pure $Sp\left(4\right)$ 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 show the first numerical evidence of a bulk phase transition in the lattice theory with dynamical Wilson fermions, and perform the technical steps necessary to set up future investigations of the mesonic spectrum of the full theory.