  We perform lattice studies of meson mass spectra and decay constants of the
$Sp(4)$ gauge theory in the quenched approximation. We consider two species of
(Dirac) fermions as matter field content, transforming in the 2-index
antisymmetric and the fundamental representation of the gauge group,
respectively. All matter fields are formulated as Wilson fermions. We
extrapolate to the continuum and massless limits, and compare to each other the
results obtained for the two species of mesons. In the case of two fundamental
and three antisymmetric fermions, the long-distance dynamics is relevant for
composite Higgs models. This is the first lattice study of this class of
theories. The global $SU(4) \times SU(6)$ symmetry is broken to the $Sp(4)
\times SO(6)$ subgroup, and the condensates align with the explicit mass terms
present in the lattice formulation of the theory. The main results of our
quenched calculations are that, with fermions in the 2-index antisymmetric
representation of the group, the masses squared and decay constant squared of
all the mesons we considered are larger than the corresponding quantities for
the fundamental representation, by factors that vary between $\sim$1.2 and
$\sim$2.7. We also present technical results that will be useful for future
lattice investigations of dynamical simulations, of composite chimera baryons,
and of the approach to large $N$ in the $Sp(2N)$ theories considered. We
briefly discuss their high-temperature behaviour, where symmetry restoration
and enhancement are expected.
