  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.
