One of the great challenges in the study of quantum matter is to understand metallic states known as non-Fermi liquid (NFL) metals, where strong correlations quickly destroy any quasiparticle excitations. Despite myriad measurements of NFL behavior in a broad range of materials, the search for organizing principles and tractable models for NFLs has proven enormously difficult. Surprisingly, I will show that exact results are in fact possible. I will show that the optical conductivity -- the current response to a time-dependent electric field -- can be exactly constrained in a class of models of Fermi surfaces near a quantum critical point. Using the emergent symmetry and quantum anomaly structure of these models, I will show that critical fluctuations of a Landau order parameter cannot generate non-trivial scaling behavior in the optical conductivity of these models. Our results suggest that the simplest kinds of Landau order parameter fluctuations cannot alone produce the optical conductivity scaling reported in some NFL materials.