# How to understand this diagram, and is it relevant to the renormalizability of a theory?

I'm having trouble understanding this diagram from my lecture note: Each grey-shaded circle represents the diagrams for the two-point function $$D(k)$$. In equation, the diagram reads$$\bar G_n(k_1,...,k_n) = \Gamma_n(k_1,...,k_n)\prod_{i=1}^nD(k_i)$$ However, I'm still unclear on how to understand this diagram and equation. Are we considering a scattering process involving $$n$$ particles?

Suppose we have a Lagrangian: $$\mathcal{L} = \frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2+\bar\psi(i\not\partial-M)\psi-g\phi\bar\psi\psi.$$ We can draw all the divergent Feynman diagrams using the superficial degree of divergence. Are those grey circles representing those divergent graphs, such as a 1-loop correction diagram? We can find that the Lagrangian also needs additional terms $$\phi^4$$ and their counterterms to be normalizable (which are not 2-point functions). Why don't we consider them in this diagram?

This diagram is meant to represent an $$n$$-point correlation function (or Green's function), $$G^{(n)}$$. It does not itself represent a scattering amplitude, but could be related to a scattering amplitude via the LSZ formula. It could also represent an internal part of a Feynman diagram for a more complicated amplitude.
The quantum effective action will involve an infinite number of terms with all powers of the fields and derivatives consistent with the symmetries. With schematic notation, $$\begin{equation} S_{\rm quantum} = \int d^4 x -\frac{Z}{2}(\partial \phi)^2 - \frac{m^2}{2} \phi^2 + \sum_{n_\phi, n_\partial} C_{n_\partial, n_\phi} \partial^{n_\partial} \phi^{n_\phi} \end{equation}$$
There are two components of this diagram. The first are the propagators with filled-in circles. These represent exact propagators and correspond to the factors $$D(k)$$ in the expression for $$G^{(n)}$$. The exact propagator can be derived from the quantum effective action by inverting the linearized equations of motion that follow from the quantum effective action.
The circle labeled $$\Gamma^i_n$$ represents an $$n$$-point vertex which follows from one of the terms in the quantum effective action with $$n_\phi=n$$.
Finally, while the expression for the Green's function is formally exact, in practice $$D$$ and $$\Gamma$$ (the propagator and vertex factor) can only be computed perturbatively.