A proof of the irreversibility of renormalization group flows in four dimensions

TL;DR

This paper proves that in four-dimensional unitary, renormalizable theories, the c-function decreases monotonically along renormalization group flows, confirming the irreversibility of these flows and extending the c-theorem to higher dimensions.

Contribution

It provides a rigorous proof of the irreversibility of RG flows in four dimensions using Ward identities and spectral methods, confirming the monotonicity of the c-function.

Findings

The c-function decreases monotonically along RG flows.

At fixed points, the c-function matches the Euler density coefficient.

Decoupling of massive modes causes the c-function's decrease away from fixed points.

Abstract

We present a proof of the irreversibility of renormalization group flows, i.e. the c-theorem for unitary, renormalizable theories in four (or generally even) dimensions. Using Ward identities for scale transformations and spectral representation arguments, we show that the c-function based on the trace of the energy-momentum tensor (originally suggested by Cardy) decreases monotonically along renormalization group trajectories. At fixed points this c-function is stationary and coincides with the coefficient of the Euler density in the trace anomaly, while away from fixed points its decrease is due to the decoupling of positive--norm massive modes.