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I'm studying chapter 28 in Srednicki (the renormalization group) and I'm having troubles figuring out how he derives eq. (28.15) (last summation above) from eqs. (28.7) and (28.9).

More specifically he states that

$$G(\alpha, \epsilon) \equiv \ln(Z_g^2Z_{\phi}^{-3})=\sum_{n=1}^{\infty}\frac{G_n(\alpha)}{\epsilon^n}\tag{28.14+15}$$

where

$$Z_{\phi}=1+\sum_{n=1}^{\infty} \frac{a_n(\alpha)}{\epsilon^n}\tag{28.7}$$

and

$$Z_g=1+\sum_{n=1}^{\infty} \frac{c_n(\alpha)}{\epsilon^n}\tag{28.9}$$

and with

$$G_1(\alpha) = 2c_1(\alpha) -3a_1(\alpha).\tag{28.16}$$

Does anybody know an identity of logarithms that I'm missing to prove this equality (28.16)?

Qmechanic
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1 Answers1

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  1. Srednicki is treating $$ Z-1\tag{A}$$ and $$\ln Z~=~-\sum_{j=1}^{\infty}\frac{(1-Z)^j}{j}\tag{B}$$ as perturbative formal power series in the coupling constant $\alpha\equiv \frac{g^2}{(4\pi)^3}$. Each coefficient of such formal power series is a truncated Laurent series in $\epsilon$. Eq. (B) seems to be the answer to OP's question.

  2. Be aware that Srednicki somewhat misleadingly writes the double sum in the opposite order. [He writes the sum over $\epsilon$-powers explicitly, while the sum over $\alpha$-powers is implicit, cf. e.g. eqs. (28.7) & (28.9).]

  3. For a deeper reason why it is consistent to perturbatively expand divergent terms, see e.g. this & this related Phys.SE posts.

Qmechanic
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