Reading "From Linear SUSY to Constrained Superfields" by Komargodski and Seiberg, I got a bit confused regarding the existence of the conserved charges in a theory with spontaneous symmetry breaking (SSB) of a global symmetry:
More precisely, in the second-to-last paragraph on page 1 we have
"When a global symmetry is spontaneously broken, the corresponding conserved charge does not exist because its correlation functions are IR divergent. However the conserved current and even the commutators with the conserved charge do exist."
I know that in the case of global SSB we have $Q|0\rangle\neq0$ for the conserved charge $Q$. However, I don't have any insight about the correlation functions. Could somehow $Q|0\rangle\neq0$ imply something like $||Q|0\rangle||=\infty$ or $\langle Q\rangle\rightarrow\infty$? And how could one see that?