Perturbative renormalizability with an infinite number of relevant and marginal operators

Abstract

We use renormalization group equations to derive conditions under which a finite number of running couplings and masses are required for perturbative renormalization, despite the appearance of an infinite number of relevant and marginal operators. These conditions are immediately relevant to the study of quantum chromodynamics in light-front field theory, where one needs to employ regulators that violate continuous symmetries such as Lorentz covariance and gauge invariance. Using simple one-loop examples, we use these conditions to show how the renormalization group itself leads naturally to O( N) symmetry in scalar theories, to massless fermions when explicit chiral invariance is broken by the regulator, and to massless gauge bosons when explicit gauge invariance is broken. We also show that these conditions may allow one to recover hidden symmetries such as φ → − φ symmetry when computing in the broken symmetry phase, which is relevant to light-front calculations without zero-modes.