Abstract : We present an example where a CFT qualitatively changes the behavior of loop diagrams at scales parametrically smaller than the mass scale where the CFT is broken. In our toy model, a large anomalous dimension leads to a scenario where the corrections to the mass of a scalar is dominated at low energies below even the scale of CFT breaking. CFTs as simple as complex Φ4 theory combined with higher dimensional operators allow for a simple realization of this effect. Surprisingly, this behavior can be seen from the IR perspective as long as higher dimensional operators are properly taken into account. In our example, multi-loop divergences can be summed into an exponential suppression yielding a convergent correction to the mass of a Yukawa coupled scalar instead of the usual quadratic divergence. Even from the bottom up, this contribution can be seen to be dominated at a scale not associated with any new particles.