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We consider the $Sp(4)$ gauge theory coupled to $N_f=2$ fundamental and$n_f=3$ antisymmetric flavours of Dirac fermions in four dimensions. Thistheory serves as the microscopic origin for composite Higgs models with$SU(4)/Sp(4)$ coset, supplemented by partial top compositeness. We studynumerically its lattice realisation, and couple the fundamental plaquetteaction to Wilson-Dirac fermions in mixed representations, by adopting a(rational) hybrid Monte Carlo method, to perform non-trivial tests of theproperties of the resulting lattice theory.  We find evidence of a surface (with boundaries) of first-order bulk phasetransitions in the three-dimensional space of bare parameters (one coupling andtwo masses). Explicit evaluation of the Dirac eigenvalues confirms the expectedpatterns of global symmetry breaking. After investigating finite volume effectsin the weak-coupling phase of the theory, for the largest available lattice westudy the mass spectra of the lightest spin-0 and spin-1 flavoured mesonscomposed of fermions in each representation, and of the lightest half-integerspin composite particle made of fermions in different representations -- thechimera baryon. This work sets the stage for future systematical studies of thenon-perturbative dynamics in phenomenologically relevant regions of parameterspace.

Lattice studies of the $Sp(4)$ gauge theory with two fundamental and three antisymmetric Dirac fermions