Regarding the propagator $\mathcal{G}(k,i\omega,r)$ of a Euclidean scalar real Gaussian quantum field theory $$\mathcal{Z_0}=\int\mathcal{D}[\phi]e^{-\mathcal{S}[\phi]}$$ $$\mathcal{S[\phi]}=\int d{\vec{q}}\int d{\omega} \, \mathcal{G}^{-1}(\vec{q}, i\omega,r) \tilde{\Phi}(\vec{q},i\omega)\tilde{\Phi}(-\vec{q},-i\omega) $$ $$\mathcal{G}(\vec q,i\omega,r)=\frac{1}{q^2+\omega^2+r} \ .$$ There is a RG fixed point at $r=0$ describing a spontaneous $\mathbb{Z}_2$ symmetry breaking. For $r>0$ the expectation value of the field $\langle\phi\rangle=0$ while for $r<0$ there is a non-zero expectation value due to the spontaneous symmetry breaking.
I have several questions regarding this framework:
- Is the propagator $\mathcal{G}(\vec q, i\omega,r)$ meaningful for $r<0$? The problem I see is that the poles of the propagator are on the real axis and that for a Gaussian FT the field is unbounded in the symmetry broken case.
- In case the propagator is meaningful in the case $r<0$. Is the propagator associated with the connected or unconnected correlation function? By connected correlation function we mean the correlation function from which the mean field value is subtracted $\langle \phi(x)\phi(y)\rangle-\langle\phi\rangle^2$ while with unconnected correlation function we refer only to the $\langle \phi(x) \phi(y) \rangle$ term.
