In recent decades, the study of many-body systems has been an active area of research in both physics and mathematics. In this talk, we will consider a system of N spin 1/2 interacting fermions confined in a box in the dilute regime, with a particular focus on the correlation energy which is defined as the difference between the ground state energy and that of the free Fermi gas. We will discuss some recent results about a first order asymptotics for the correlation energy in the thermodynamic limit where the number of particles and the size of the box are sent to infinity keeping the density fixed. In particular, we will present a new upper bound for the correlation energy, which is consistent with the well-known Huang-Yang formula from 1957.