Field independent additive constant in Wilson actions

We discuss the field independent additive constant in Wilson actions carefully within the exact renormalization group formalism. The additive constant does not affect the correlation functions of fields normalized by the partition function, and for that reason it is often ignored. But it is an essential part of the partition function, and in the limit where the UV cutoff goes to zero, the constant gives a renormalized vacuum energy density. We discuss some concrete examples: the massless and massive Gaussian theory for a single component scalar field and a Dirac fermion and the linear sigma model in the large $N$ limit.