Soft Theorems and Dilaton Effective Theory

We derive a new model-independent double-soft dilaton theorem, taking into account the spacetime dependence of the dilation commutator $[i Q_D,{\cal O}(x)]= (\Delta_{\cal O} + x \cdot \partial){\cal O}(x)$. The procedure restores positivity in the (pseudo)-Goldstone masses and sets the constraint $\Delta_{\cal O} = d-2\,$ for a single operator ${\cal O}$ responsible for generating a dilaton mass.We discuss gravitational form factors as a tool to probe infrared conformality in field theories with particle content. In a second part we explore to what extent QCD-like gauge theories (in the chiral limit) could fit into this category. We find that the quark bilinear has scaling dimension $\Delta_{\bar qq} = d-2$, therefore satisfying the double-soft theorem. We show that some findings are realised in ${\cal N}=1$ supersymmetric gauge theories and argue that the extension below the conformal window makes sense in that case.