Exploring the conformal transition from above and below

We consider conformal transitions arising from the merging of IR and UV fixed points, expected to occur in QCD with a large enough number of flavors. We study the smoothness of physical quantities across this transition, being mostly determined by the logarithmic breaking of conformal invariance. We investigate this explicitly using holography where approaching the conformal transition either from outside or inside the conformal window (perturbed by a mass term) is characterized by the same dynamics. The mass of spin-1 mesons and $F_\pi$ are shown to be continuous across the transition, as well as the dilaton mass. This implies that the lightness of the dilaton cannot be a consequence of the spontaneous breaking of scale invariance when leaving the conformal window. Our analysis suggests that the light scalar observed in QCD lattice simulations is a $q\bar q$ meson that becomes light since the $q\bar q$-operator dimension reaches its minimal value.