Resummation of single class of diagrams vs all 1PI diagrams

Question

Maggiore considers on page 136 in Section 5.6 Renormalization in the book A Modern Introduction to Quantum Field Theory, the resummation of tadpole diagrams as its own individual geometric series to give

$$\frac{i}{p^2-m^2-B}\tag{5.109}\label{1}$$

where $-iB$ is the contribution from a single tadpole diagram (see Fig. 5.11). This makes sense and is done by many books in order to find the renormalized mass to first order. However, he then does the same with the two-loop 'Saturn diagram' (see Fig. 5.13), again summing it as its own geometric series. Am I right in saying that it doesn't make sense to include Saturn diagrams up to all orders, but not the tadpole diagram at all, as a single Saturn diagram is of the same order as two tadpole diagrams?

For example, most books don't seem to do this but rather perform the resummation of all 1-particle irreducible diagrams $-i\Sigma(p)$ getting

$$\frac{i}{p^2-m^2-\Sigma(p)}\tag{2}$$

which gives back Eq. (\ref{1}) to first order.

Answer 1

Yes, OP has a point. On p. 130-131 and p .136 Maggiore is considering the connected 2-point function $G_c$ in $\lambda\phi^4$ theory, but he forgets to inform the reader that his counting of coupling constants $\lambda$ and loops refer to the self-energy $\Sigma$.

So e.g. the tadpole diagram Fig. 5.11 (which is more correctly called a self-loop diagram) and the Saturn/sunset diagram Fig. 5.13 contribute to $\Sigma$ at 1-loop and 2-loop, respectively.

OP already seems aware of the fact that $G_c$, $G_0$ and $\Sigma$ are connected via a geometric series, cf. e.g. this Phys.SE post and OP's eq. (2).