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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 $Y\varphi$ 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 $$\mathcal L_c=-\frac12(Z_\varphi-1)\partial^\mu\varphi\partial_\mu\varphi-\frac12(Z_m-1)m^2\varphi^2,$$ 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?

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$Y$ 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 $\langle 0|\phi(x)|0\rangle = 0$. 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.

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  • $\begingroup$ Thanks for your reply. Did Srednicki also forget figure 9.3 which also contributes to $O(g^3)$ term of the VEV? By the way, the $\mathcal L_c$ courterterm is included not to ensure the vanish of the VEV but to absorb the infinity of the loop corrections to the propagator in figure 14.1, the VEV can be made zero by $Y\varphi$ whether $\mathcal L_c$ is included or not. $\endgroup$ – Shin-Yue Jun 6 '17 at 12:11
  • $\begingroup$ 1. The complete set of diagrams needed to contribute the $O(g^3)$ terms is very large, and I think Srednicki did not aim to list the complete set in this chapter because renormalization beyond the leading order is not important at this stage. I do not check whether his list is complete, but counter terms must contribute to this list. If you want to know more about renormalization beyond the leading order, section 11.4 and 11.5 of Peskin's book will be fine. $\endgroup$ – Eric Yang Jun 8 '17 at 4:54
  • $\begingroup$ 2. That is my abuse of language, sorry. $\mathcal{L}_{c}$ courter term is included to absorb the infinity of propagator and three point function. What I mean is when you calculate the VEV of the theory, $\mathcal{L}_{c}$ and $Y\phi$ must be both included to make VEV = 0. If $\mathcal{L}_{c}$ is not included. You can not say VEV = 0. $\endgroup$ – Eric Yang Jun 8 '17 at 5:08

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