  We consider a linear sigma model describing $2N_f^2$ bosons ($\sigma$, ${\bf
a_0}$, $\eta '$ and ${\bf \pi}$) as an approximate effective theory for a
$SU(3)$ local gauge theory with $N_f$ Dirac fermions in the fundamental
representation. The model has a renormalizable $U(N_f)_L\bigotimes U(N_f)_R$
invariant part, which has an approximate $O(2N_f^2)$ symmetry, and two
additional terms, one describing the effects of a $SU(N_f)_V$ invariant mass
term and the other the effects of the axial anomaly. We calculate the spectrum
for arbitrary $N_f$. Using preliminary and published lattice results from the
LatKMI collaboration, we found combinations of the masses that vary slowly with
the explicit chiral symmetry breaking and $N_f$. This suggests that the anomaly
term plays a leading role in the mass spectrum and that simple formulas such as
$M_\sigma^2\simeq (2/N_f-C_\sigma)M_{\eta '}^2$ should apply in the chiral
limit. Lattice measurements of $M_{\eta '}^2$ and of approximate constants such
as $C_\sigma$ could help locating the boundary of the conformal window. We show
that our calculation can be adapted for arbitrary representations of the gauge
group and in particular to the minimal model with two sextets, where similar
patterns are likely to apply.
