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 $[i{Q}_D,\mathcal{O}(x)]=({\mathrm{\Delta }}_{\mathcal{O}}+x\cdot \mathrm{\partial })\mathcal{O}(x)$. The procedure restores positivity in the (pseudo)-Goldstone masses and sets the constraint ${\mathrm{\Delta }}_{\mathcal{O}}=d-2$ for a single operator $\mathcal{O}$ 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 ${\mathrm{\Delta }}_{\bar{q}q}=d-2$, therefore satisfying the double-soft theorem. We show that some findings are realised in $\mathcal{N}=1$ supersymmetric gauge theories and argue that the extension below the conformal window makes sense in that case.