Irrelevant deformations and the holographic Callan-Symanzik equation

TL;DR

This paper develops a general framework for deriving the Callan-Symanzik equation in holography, incorporating new renormalization techniques, and explores the effects of multi-trace operators and anomalies in conformal field theories.

Contribution

It provides a comprehensive formula for the holographic Callan-Symanzik equation, including effects of multi-trace operators and recent holographic renormalization results.

Findings

New logarithmic terms lead to non-trivial beta functions.

Multi-trace counterterms cause specific non-linearities in the equation.

Computed conformal anomaly for a scalar three-point function.

Abstract

We discuss the systematics of obtaining the Callan-Symanzik equation within the framework of the gauge/gravity dualities. We present a completely general formula which in particular takes into account the new holographic renormalization results of arXiv:1102.2239. Non-trivial beta functions are obtained from new logarithmic terms in the radial expansion of the fields. The appearance of multi-trace counterterms is also discussed in detail and we show that mixing between single- and multi-trace operators leads to very specific non-linearities in the Callan-Symanzik equation. Additionally, we compute the conformal anomaly for a scalar three-point function in a CFT.