Irrelevant deformations and the holographic Callan-Symanzik equation

TL;DR

The paper develops a comprehensive holographic framework for deriving the Callan-Symanzik equation in gauge/gravity duality, explicitly incorporating irrelevant deformations, logarithmic divergences, and the emergence of multi-trace counterterms. Through a detailed analysis of a scalar toy model in AdS$_3$ with a cubic interaction, it derives the holographic CS equation, computes conformal anomalies for two- and three-point functions, and demonstrates how beta functions and operator mixing arise from bulk renormalization data. The work obtains explicit all-order structures (where possible) and provides concrete formulas for anomalies and beta functions, including a new conformal anomaly for a scalar three-point function and a precise relationship between $c_3$, $c_2$, and the bulk coupling $\lambda$. The approach unifies the holographic and field-theoretic perspectives on renormalization, offering a general method applicable to multiple bulk fields and prompting further exploration of non-AlAdS spacetimes, Weyl rescalings, and finite-cutoff holography.

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.