Decoupling of High Dimension Operators from the Low Energy Sector in Holographic Models

TL;DR

This work analyzes how high-dimension operators decouple from the low-energy sector in broken CFTs using holographic duals. It reveals two decoupling mechanisms: in hard-wall (RS-type) models, high bulk masses imply large 4D masses, yielding conventional decoupling; in soft-wall models, bulk locality combined with the Schrödinger potential produces exponential suppression of overlaps between heavy-dimension operators and light states, often depending on operator dimension as $e^{-\\lambda \Delta^p}$. The authors introduce a two-point form factor $f(\Delta_1,\Delta_2)$ capturing cross-dimension couplings and demonstrate how NEC constraints shape viable soft-wall backgrounds. The results justify holographic EFTs for QCD and condensed matter by showing that high-dimension operators contribute negligibly to low-energy observables, thus supporting bulk light degrees of freedom as the dominant mediators of low-energy dynamics.

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.

Figures (1)

• Figure 1: The Schrodinger potential in equation (\ref{['eq:schr']}) for the KK mode wavefunctions gets contributions from the bulk mass as well as from the deviations from AdS due to the soft wall in the IR. The former dominates at small $z$, behaving like $\frac{\Delta^2}{z^2}$ and pushing the wavefunctions toward larger $z$, whereas the latter tend to push the wavefunctions toward smaller $z$. We have shown an example where the dilaton profile $\Phi(z)$ in equation (\ref{['eq:SeparationVariables']}) behaves like $\sim z^2$, resulting in a potential $\propto z^2$ at large $z$, but the pattern is general. Modes are contained in a finite-sized 'cavity' in the bulk, with a central $z$ that tends to grow with $\Delta$.