Soft Theorems and Dilaton Effective Theory

We discuss model independent single- and double-soft dilaton theorems, 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 an operator ${\cal O}$ generating a dilaton mass. We then apply these findings to QCD-like gauge theories where it has been speculated that a dilaton phase might emerge in the chiral limit. It is found that the quark bilinear is of scaling dimension $\Delta_{\bar qq} = d-2$, therefore satisfying the 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. We briefly discuss the infrared implementation of conformal symmetry by the example of gravitational form factors.