I am trying to calculate the beta function of the 2D Gross-Neveu model after performing a Hubbard-Stratanovich transformation. Of course, you can calculate it without this transformation, but I am doing this as a warm-up for something else.
My initial Lagrangian is
\begin{equation} \mathcal{L}=\bar{\psi}_i(i\partial\kern-0.3em\raise0.1ex\hbox{/}-m)\psi_i+\frac{1}{2}g^2(\bar{\psi}_i\psi_i)^2 \end{equation}
and I integrate in a new field $\phi$ to get
\begin{equation} \mathcal{L}_{\phi}=\bar{\psi}_i(i\partial\kern-0.3em\raise0.1ex\hbox{/}-m)\psi_i+\frac{1}{2}\phi^2+ig\phi\bar{\psi}_i\psi_i \end{equation}
I am confused about the Feynman rules for $\phi$ since there is no kinetic term. I see in a comment here that perhaps there are no internal lines at all for $\phi$. But then there are no loop diagrams to calculate and I don't know how to do the usual procedure to find counterterms.
Even if there actually are internal lines for $\phi$, I don't see how they could have momentum dependence. My naive guess is that the propagator would just be a constant like $i$ in Minkowski signature.
So, how do the Feynman diagrams and rules work out for this scalar with no kinetic term?