[Paper Review] Decoupling of High Dimension Operators from the Low Energy Sector in Holographic Models

A. Liam Fitzpatrick, Jared Kaplan+2 more|arXiv (Cornell University)|Apr 11, 2013

Black Holes and Theoretical Physics52 references13 citations

TL;DR

This paper demonstrates that high-dimension operators in broken conformal field theories (CFTs) decouple rapidly from low-energy physics—especially mesons and glueballs—due to exponential suppression in their coupling to light states. Using holographic duality, it shows this decoupling arises from the wavefunction localization of high-dimension operators in the bulk AdS space, where their overlap with light Kaluza-Klein modes diminishes exponentially with operator dimension, ensuring consistency between d-dimensional EFT and (d+1)-dimensional holographic descriptions.

ABSTRACT

We study the decoupling of high dimension operators from the the description of the low-energy spectrum in theories where conformal symmetry is broken by a single scale, which we refer to as `broken CFTs'. Holographic duality suggests that this decoupling occurs in generic backgrounds. We show how the decoupling of high mass states in the (d+1)-dimensional bulk relates to the decoupling of high energy states in the d-dimensional broken CFT. In other words, we explain why both high dimension operators and high mass states in the CFT decouple from the low-energy physics of the mesons and glueballs. In many cases, the decoupling can occur exponentially fast in the dimension of the operator. Holography motivates a new kind of form factor proportional to the two point function between broken CFT operators with very different scaling dimensions. This new notion of decoupling can provide a systematic justification for holographic descriptions of QCD and condensed matter systems with only light degrees of freedom in the bulk.

Motivation & Objective

• To understand the decoupling of high-dimension operators from low-energy spectra in broken CFTs, particularly in holographic models.
• To reconcile the d-dimensional EFT description of light mesons and glueballs with the (d+1)-dimensional bulk EFT in warped AdS space.
• To establish that the decoupling of high-dimension operators is not merely power-law but can be exponential, ensuring compatibility between bulk and boundary EFTs.
• To provide a systematic justification for holographic models of QCD and condensed matter systems that include only light degrees of freedom in the bulk.

Proposed method

• Using holographic duality, the authors map the conformal Casimir eigenvalues in the d-dimensional CFT to the mass eigenvalues of bulk fields in (d+1)-dimensional AdS space.
• They analyze the overlap between bulk Kaluza-Klein modes and boundary operators of varying scaling dimensions, focusing on the wavefunction behavior in the small-z (UV) limit.
• The key equation relates the boundary operator overlap coefficient $ f_{\mathcal{O}_2,0} $ to the bulk wavefunction $ g_{02}(z) $, showing that large $ \Delta_2 $ leads to rapid suppression via Bessel function asymptotics.
• In hard-wall models, the wavefunction for high-dimension operators scales as $ z^{\Delta_2 - 3/2} $, leading to exponentially small normalization and thus decoupling.
• The normalization condition ensures that only the lightest mode contributes significantly to the norm, forcing the overlap coefficient to scale as $ \Gamma(\Delta_2 - 1)^{-1} $, which decays exponentially with $ \Delta_2 $.
• The analysis is extended to fermionic and scalar fields, showing consistent suppression across different spin sectors.

Experimental results

Research questions

• RQ1 How does the coupling of high-dimension operators to light mesons and glueballs behave in a broken CFT with a mass gap?
• RQ2 What is the functional dependence of the form factor $ f(\Delta_1, \Delta_2) $ on the scaling dimension $ \Delta_2 $ when $ \Delta_2 \to \infty $?
• RQ3 Why is the decoupling of high-dimension operators from low-energy physics in holographic models not just power-law but often exponential?
• RQ4 How does the wavefunction of a bulk Kaluza-Klein mode with large dual operator dimension behave in the UV region, and what does this imply for its overlap with light states?
• RQ5 Can the decoupling in the boundary CFT be systematically linked to the decoupling of heavy bulk states in the holographic dual?

Key findings

• The form factor $ f_{\Delta_1}(\Delta) $ between a low-dimension operator and a high-dimension operator decays exponentially with $ \Delta $, specifically as $ \exp[-\lambda \Delta^p] $, with $ p=1 $ in soft-wall AdS/QCD models.
• In hard-wall models, the overlap coefficient $ f_{\mathcal{O}_2,0} $ scales as $ \Gamma(\Delta_2 - 1)^{-1} $, leading to exponential suppression with increasing $ \Delta_2 $, even when bulk parameters are tuned.
• The wavefunction of a high-dimension operator in the bulk is dominated by its small-$ z $ behavior, $ g_{02}(z) \propto z^{\Delta_2 - 3/2} $, which suppresses its contribution to the norm and coupling to light modes.
• The normalization of the lightest Kaluza-Klein mode is dominated by the light operator’s component, so the high-dimension operator’s contribution becomes negligible as $ \Delta_2 \to \infty $.
• The decoupling is robust: even when bulk coupling parameters are tuned to maintain constant mass and boundary coupling, the wavefunction localization still causes exponential suppression.
• The result ensures consistency between the d-dimensional EFT (based on particle masses) and the (d+1)-dimensional EFT (based on bulk masses), validating holographic descriptions of QCD and condensed matter systems with only light bulk fields.