  We discuss the extension of the field theoretical approach, already used in
the lattice determination of the topological susceptibility, to the computation
of further terms in the expansion of the ground state energy $F(\theta)$ around
$\theta = 0$ in SU(N) Yang-Mills theories. In particular we determine the
fourth order term in the expansion for SU(3) pure gauge theory and compare our
results with previous cooling determinations. In the last part of the paper we
make some considerations about the nature of the ultraviolet fluctuations
responsible for the renormalization of the lattice topological charge
correlation functions; in particular we propose and test an ansatz which leads
to improved estimates of the fourth and higher order terms in the expansion of
F(\theta).
