# Shift in renormalized Green's function

In chapter 12.2 p. 410 of Peskin and Schroeder the Callan-Symanzik equation is derived. I understand the relation between (connected) renormalized and non-renormalized Green's functions given by $$G^{(n)}(x_1,\ldots,x_n)=Z^{-n/2}G_0^{(n)}(x_1,\ldots,x_n)$$ where $$Z$$ is given by the field strength renormalization $$\phi=Z^{-1/2}\phi_0$$ and then with a shift in our renormalization scale $$M\rightarrow M+\delta M$$ we have a rescaling of $$\lambda\rightarrow \lambda + \delta\lambda$$ and $$\phi\rightarrow (1+\delta\eta)\phi$$ where we notice that this definition gives $$Z^{1/2}=(1-\delta\eta)$$ What I don't understand is how this gives a shift in $$G^{(n)}$$ which is given by the equation below 12.37 of the book, namely $$G^{(n)}\rightarrow G^{(n)}(1+n\delta\eta)$$ Notice that the coefficient of $$\delta\eta$$ is $$n$$ (power of fields) and not $$\eta$$.

It is assumed that the bare quantities do not depend on the renormalization scale, i.e. if $$M\rightarrow M + \delta M$$, $$\lambda \rightarrow \lambda + \delta\lambda$$ and $$\phi \rightarrow (1+\delta\eta)\phi$$ they do not change. In particular the bare Greensfunction $$G_0$$ and $$\phi_0$$ do not change.

We observe that

$$\phi \sim \frac{1}{Z^{1/2}}$$

therefore if $$\phi \rightarrow (1+\delta\eta)\phi$$ then we get:

$$Z^{1/2} \rightarrow Z^{1/2} (1- \delta\eta) \quad\text{but also} \quad Z^{-1/2} \rightarrow Z^{-1/2} (1+ \delta\eta)$$

as mentioned in the post.

However, $$G^{(n)}(x_1,\ldots,x_n) \sim (Z^{-1/2})^n$$, remember the bare Greensfunction does not depend on the renormalization scaling. Therefore

$$G^{(n)}(x_1,\ldots,x_n) \rightarrow G^{(n)}(x_1,\ldots,x_n)(1+ \delta\eta)^n \approx G^{(n)}(x_1,\ldots,x_n)(1+n\delta\eta + O(\delta\eta^2) )$$

or shortly:

$$G^{(n)}(x_1,\ldots,x_n) \rightarrow G^{(n)}(x_1,\ldots,x_n)(1+n\delta\eta)$$

Actually $$G^{(n)}$$ is the $$n$$-point function, i.e. is the vacuum expectance value of $$n$$ field operators. If 1 field operator scales like $$\phi \rightarrow (1+\delta\eta)\phi$$, $$n$$ of those scale like

$$\phi^n \rightarrow (1+n\delta\eta)\phi^n$$.