The Constraints of Conformal Symmetry on RG Flows

TL;DR

This paper explores how conformal symmetry constrains renormalization group flows in quantum field theory, providing new proofs of key theorems and explicit calculations in four-dimensional theories.

Contribution

It offers a novel proof of Zamolodchikov's theorem and derives the a-theorem using conformal symmetry constraints, with explicit examples in four-dimensional theories.

Findings

New proof of Zamolodchikov's theorem

Derivation of the a-theorem from conformal symmetry

Explicit calculations of the a-anomaly in specific models

Abstract

If the coupling constants in QFT are promoted to functions of space-time, the dependence of the path integral on these couplings is highly constrained by conformal symmetry. We begin the present note by showing that this idea leads to a new proof of Zamolodchikov's theorem. We then review how this simple observation also leads to a derivation of the a-theorem. We exemplify the general procedure in some interacting theories in four space-time dimensions. We concentrate on Banks-Zaks and weakly relevant flows, which can be controlled by ordinary and conformal perturbation theories, respectively. We compute explicitly the dependence of the path integral on the coupling constants and extract the change in the a-anomaly (this agrees with more conventional computations of the same quantity). We also discuss some general properties of the sum rule found in arXiv:1107.3987 and study it in several examples.