The proton radius (puzzle?) and its relatives

Abstract

We review determinations of the electric proton charge radius from a diverse set of low-energy observables. We explore under which conditions it can be related to Wilson coefficients of appropriate effective field theories. This discussion is generalized to other low-energy constants. This provides us with a unified framework to deal with a set of low-energy constants of the proton associated with its electromagnetic interactions. Unambiguous definitions of these objects are given, as well as their relation with expectation values of QCD operators. We show that the proton radius obtained from spectroscopy and lepton–proton scattering (when both the lepton and proton move with nonrelativistic velocities) is related to the same object of the underlying field theory with $O\left(\alpha \right)$ precision. The model dependence of these analyses is discussed. The prospects of constructing effective field theories valid for the kinematic configuration of present, or near-future, lepton–proton scattering experiments are explored.