  The scaling laws in an infrared conformal (IR) theory are dictated by the
critical exponents of relevant operators. We have investigated these scaling
laws at leading order in two previous papers. In this work we investigate
further consequences of the scaling laws, trying to identify potential
signatures that could be studied by lattice simulations. From the first
derivative of the form factor we derive the behaviour of the mean charge radius
of the hadronic states in the theory. We obtain $\langle r_H^2 \rangle \sim
m^{-2/(1+\gamma^*_m)}$ which is consistent with $\langle r_H^2 \rangle \sim
1/M_H^{2}$. The mean charge radius can be used as an alternative observable to
assess the size of the physical states, and hence finite size effects, in
numerical simulations. Furthermore, we discuss the behaviour of specific field
correlators in coordinate space for the case of conformal, scale-invariant, and
confining theories making use of selection rules in scaling dimensions and
spin. We compute the scaling corrections to correlations functions by
linearizing the renormalization group equations. We find that these correction
are potentially large close to the edge of the conformal window. As an
application we compute the scaling correction to the formula $M_H \sim
m^{1/(1+\gamma_m^*)}$ directly through its associated correlator as well as
through the trace anomaly. The two computations are shown to be equivalent
through a generalisation of the Feynman-Hellmann theorem for the fermion mass,
and the gauge coupling.
