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

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