Multifractals, operator-product expansion, and field theory

We explore possible distinctions between multifractal scaling phenomena and Lagrangian field theories (FT’s) describing standard critical phenomena, via the operator-product expansion. While the scaling dimensions ${\mathit{x}}_{\mathit{n}}$ of multifractal moments must be convex functions of the order n, analogous FT exponents of powers of the field are concave, by stability and correlation inequalities, and cannot describe multifractal scaling. However, powers of gradients of the field may lead to a novel and unexpected multifractal convexity in a FT, such as, e.g., the nonlinear σ model.