Higher order correction of the VEV of the phi-cubed theory in Srednicki Chapter 9

Question

In Srednicki's QFT Chapter 9, he first computed the vacuum expectation value of the field $varphi(x)$ without including the counterterms in $mathcal L$, then he found that the VEV is not zero, so he included the linear counterterm $Yvarphi$ to cancel the nonzero terms in the VEV. He computed the $O(g)$ term in $Y$ and said the $O(g^3)$ term in $Y$ can also be determined if we sum up the corresponding diagrams at $O(g^3)$.
Here I have a question, he seemed to have ignored the single source diagrams which contain the vertex corresponding to the other counterterm $$mathcal L_c=-frac12(Z_varphi-1)partial^muvarphipartial_muvarphi-frac12(Z_m-1)m^2varphi^2,$$ diagrams containing this vertex do not appear at $O(g)$ in the VEV, so the $O(g)$ term in $Y$ doesn't change, but new diagrams containing this new vertex appear at $O(g^3)$ in the VEV, so the $O(g^3)$ term in $Y$ will change if we include the new vertex. Is Srednicki wrong for ignoring the effect of this vertex on the VEV?

Eric Yang · Answer

$Y$ is a function of $g$.
We can expand it in series of $g$,i.e.
$$Y(g) = y_1 g + y_3 g^3 + cdots$$
When we want to determine the value of $y_1$, the counterterms can be neglected because  it is of order $O(g^3)$.
But when we want to determine the value of $y_3$ and higher order terms, diagrams with counterterms must be included to ensure the higher order terms of $rm VEV$ vanish.
And $Y(g)$ can be calculated order by order. That is the so called perturbation quantum field theory.

Srednicki's book says,

Thus, at $O(g^3)$, we sum up the diagrams of figs. 9.4 and 9.12, and then add to $Y$ whatever $O(g^3)$ term is needed to maintain $langle 0|phi(x)|0rangle = 0$. In this way we can determine the value of $Y$ order by order in powers of $g$.

Fig9.4 and 9.12 do not include diagram with counterterms. So it may be a negligence of the author.

Higher order correction of the VEV of the phi-cubed theory in Srednicki Chapter 9

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