On Renormalization Group Flows in Four Dimensions

TL;DR

This paper explores renormalization group flows in four dimensions, linking them to spontaneously broken conformal symmetry, and establishes the a-theorem through dilaton scattering and anomaly analysis.

Contribution

It provides a detailed analysis of trace anomalies and their implications for the effective action of the Nambu-Goldstone boson, leading to a proof of the a-theorem.

Findings

The a-anomaly contributes positively to the dilaton scattering amplitude.

Unitarity implies a monotonically decreasing function between UV and IR fixed points.

The a-theorem is established via the behavior of the dilaton S-matrix.

Abstract

We discuss some general aspects of renormalization group flows in four dimensions. Every such flow can be reinterpreted in terms of a spontaneously broken conformal symmetry. We analyze in detail the consequences of trace anomalies for the effective action of the Nambu-Goldstone boson of broken conformal symmetry. While the c-anomaly is algebraically trivial, the a-anomaly is "non-Abelian," and leads to a positive-definite universal contribution to the S-matrix of 2->2 dilaton scattering. Unitarity of the S-matrix results in a monotonically decreasing function that interpolates between the Euler anomalies in the ultraviolet and the infrared, thereby establishing the a-theorem.