  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 two concrete examples: the Gaussian theory and the linear sigma model
in the large $N$ limit.
