  We present a detailed analytic study on the three-dimensional sphere of the
most popular candidates for $C$-functions, both for Dirac and scalar free
massive fields. We discuss to which extent the effective action, the R\'enyi
entanglement entropy and the renormalized entanglement entropy fulfill the
conditions expected from $C$-functions. In view of the absence of a good
candidate in the case of the scalar field, we introduce a new candidate, which
we call the modified effective action, and analyze its pros and cons.
