Naturally light dilatons from nearly marginal deformations

We discuss the presence of a light dilaton in CFTs deformed by a nearly-marginal operator O, in the holographic realizations consisting of confining RG flows that end on a soft wall. Generically, the deformations induce a condensate <O>, and the dilaton mode can be identified as the fluctuation of <O>. We obtain a mass formula for the dilaton as a certain average along the RG flow. The dilaton is naturally light whenever i) confinement is reached fast enough (such as via the condensation of O) and ii) the beta function is small (walking) at the condensation scale. These conditions are satisfied for a class of models with a bulk pseudo-Goldstone boson whose potential is nearly flat at small field and exponential at large field values. Thus, the recent observation by Contino, Pomarol and Rattazzi holds in CFTs with a single nearly-marginal operator. We also discuss the holographic method to compute the condensate <O>, based on solving the first-order nonlinear differential equation that the beta function satisfies.