  Recently, we developed an effective theory of pions and a light dilatonic
meson for gauge theories with spontaneously broken chiral symmetry that are
close to the conformal window. The pion mass term in this effective theory
depends on an exponent $y$. We derive the transformation properties under
dilatations of the renormalized fermion mass, and use this to rederive
$y=3-\gamma_m^*$, where $\gamma_m^*$ is fixed-point value of the mass anomalous
dimension at the sill of the conformal window. This value for $y$ is consistent
with the trace anomaly of the underlying near-conformal gauge theory.
