  We perform a systematic study of the modifications to the QCD vacuum energy
density $\epsilon_{vac}$ in the zero-temperature case ($T=0$) caused by a
small, but non-zero, value of the parameter $\theta$, using different effective
Lagrangian models which include the flavour-singlet meson field and implement
the $U(1)$ axial anomaly of the fundamental theory. In particular, we derive
the expressions for the topological susceptibility $\chi$ and for the second
cumulant $c_4$ starting from the $\theta$ dependence of
$\epsilon_{vac}(\theta)$ in the various models that we have considered.
Moreover, we evaluate numerically our results, so as to compare them with each
other, with the predictions of the Chiral Effective Lagrangian, and, finally,
also with the available lattice data.
