  We consider a vector-like gauge theory of fermions that confines at the
multi-TeV scale, and that realizes the Higgs particle as a composite Goldstone
boson. The weak interactions are embedded in the unbroken subgroup $Sp(4)$ of a
spontaneously broken $SU(4)$ flavour group. The meson resonances appear as
poles in the two-point correlators of fermion bilinears, and include the
Goldstone bosons plus a massive pseudoscalar $\eta'$, as well as scalars,
vectors and axial vectors. We compute the mass spectrum of these mesons, as
well as their decay constants, in the chiral limit, in the approximation where
the hypercolour $Sp(2N)$ dynamics is described by four-fermion operators, \`a
la Nambu-Jona Lasinio. By resumming the leading diagrams in the $1/N$
expansion, we find that the spin-one states lie beyond the LHC reach, while
spin-zero electroweak-singlet states may be as light as the Goldstone-boson
decay constant, $f\sim 1$ TeV. We also confront our results with a set of
available spectral sum rules. In order to supply composite top-quark partners,
the theory contains additional fermions carrying both hypercolour and ordinary
colour, with an associated flavour symmetry-breaking pattern $SU(6)/SO(6)$. We
identify and analyse several non-trivial features of the complete two-sector
gauge theory: the 't~Hooft anomaly matching conditions; the higher-dimension
operator which incorporates the effects of the hypercolour axial-singlet
anomaly; the coupled mass-gap equations; the mixing between the singlet mesons
of the two sectors, resulting in an extra Goldstone boson $\eta_0$, and novel
spectral sum rules. Assuming that the strength of the four-fermion interaction
is the same in the two sectors, we find that the coloured vector and scalar
mesons have masses $\gtrsim 4 f$, while the masses of coloured pseudo-Goldstone
bosons, induced by gluon loops, are $\gtrsim 1.5f$.
