The scaling laws in an 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 behavior of the mean charge radius of the hadronic states in the theory. We obtain $⟨r_H^2⟩\sim m^{-2/(1+{\gamma }_m^{*})}$ which is consistent with $⟨r_H^2⟩\sim 1/M_H^2$. 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 behavior 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 corrections 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 generalization of the Feynman-Hellmann theorem for the fermion mass and the gauge coupling.

• Received 3 July 2013

DOI:https://doi.org/10.1103/PhysRevD.89.014503