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The scaling laws in an infrared conformal (IR) theory are dictated by thecritical exponents of relevant operators. We have investigated these scalinglaws at leading order in two previous papers. In this work we investigatefurther consequences of the scaling laws, trying to identify potentialsignatures that could be studied by lattice simulations. From the firstderivative of the form factor we derive the behaviour of the mean charge radiusof the hadronic states in the theory. We obtain $\langle r_H^2 \rangle \simm^{-2/(1+\gamma^*_m)}$ which is consistent with $\langle r_H^2 \rangle \sim1/M_H^{2}$. The mean charge radius can be used as an alternative observable toassess the size of the physical states, and hence finite size effects, innumerical simulations. Furthermore, we discuss the behaviour of specific fieldcorrelators in coordinate space for the case of conformal, scale-invariant, andconfining theories making use of selection rules in scaling dimensions andspin. We compute the scaling corrections to correlations functions bylinearizing the renormalization group equations. We find that these correctionare potentially large close to the edge of the conformal window. As anapplication we compute the scaling correction to the formula $M_H \simm^{1/(1+\gamma_m^*)}$ directly through its associated correlator as well asthrough the trace anomaly. The two computations are shown to be equivalentthrough a generalisation of the Feynman-Hellmann theorem for the fermion mass,and the gauge coupling.

Conformal scaling and the size of $m$-hadrons