The hierarchy problem can be represented as a tension between the need for a large cut-off scale suggested, for instance, by flavor physics and the need for a low cut-off scale suggested by naturalness in electroweak symmetry breaking. I will illustrate how this tension could be largely alleviated if the Standard Model flowed to an approximate CFT above the weak scale with a specific relation among the scaling dimensions of the Higgs sector fields. To investigate the viability of that scenario one is led to ask the following simple question: in an arbitrary CFT, given a scalar operator phi, and the operator S=phi phi defined as the lowest dimension scalar S which appears in the OPE phi phi, what is the bound (that is d(S) is smaller than f(d(phi))) on the scaling dimensions of the two operators? I will present a derivation of the bound based on general considerations of OPE, conformal block decomposition, and crossing symmetry. The function f(d(phi)) is computed numerically. When d(phi) goes to 1, one has f(d(phi))=2+O(sqrt(d(phi)-1)), which shows that the free theory limit is approached continuously. An analogous bound can be derived in 2D where some non-trivial consistency check can be made. I will discuss the relevance of the result for the hierarchy problem and illustrate the directions of future investigation.