  We consider a system of $N\gg 1$ interacting fermionic particles in three
dimensions, confined in a periodic box of volume $1$, in the mean-field
scaling. We assume that the interaction potential is bounded and small enough.
We prove upper and lower bounds for the correlation energy, which are optimal
in their $N$-dependence. Moreover, we compute the correlation energy at leading
order in the interaction potential, recovering the prediction of second order
perturbation theory. The proof is based on the combination of methods recently
introduced for the study of fermionic many-body quantum dynamics together with
a rigorous version of second-order perturbation theory, developed in the
context of non-relativistic QED.
