Conformal scaling and the size of $m$-hadrons

TL;DR

This paper explores the consequences of conformal scaling laws in IR theories, focusing on hadron size, correlators, and scaling corrections, providing potential signatures for lattice simulations and insights into conformal window effects.

Contribution

It derives the behavior of hadron charge radii from form factors, analyzes correlator scaling corrections, and links these to mass formulas and the conformal window, extending previous leading-order studies.

Findings

Mean charge radius scales as m^{-2/(1+γ*_m)}

Scaling corrections to correlators can be large near the conformal window edge

Two methods for computing mass scaling corrections are shown to be equivalent

Abstract

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.