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What is the meaning of the expansion at first order ${\cal O}(\alpha_s)$ in $\delta_2$ and $\delta_3$ at the second step in the last line? These quantities are not "small" - on the contrary, the entire point is to then take the $\epsilon \to 0$ limit and the counterterms blow up.

Qmechanic
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Siupa
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1 Answers1

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The brief answer is that renormalization is first-and-foremost a perturbative formal power series in the coupling constant $\alpha_s$. E.g. a $Z$-factor is a formal power series $$Z~=~ \sum_{n=0}^{\infty} \alpha_s^nZ_n, \qquad Z_{n=0}~=~1.\tag{A}$$ Secondly, each coefficient $$Z_n=\sum_{m=-N}^{\infty}\epsilon^m Z_{nm}\tag{B}$$ of this formal power series is a truncated Laurent series in $\epsilon$. The coefficients are not necessarily small, as OP already has observed.

Eqs. (77)-(80) consider in particular the first-order coefficient $Z_{n=1}$.

See also e.g. this related Phys.SE post.

Qmechanic
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  • Doesn't the formal power series make sense only if we sum all the terms? Here we are expanding $1/Z_2$ and $1/\sqrt(Z_3)$ in $\delta_2$ and $\delta_3$ only at first order – Siupa Feb 08 '23 at 23:46
  • The sum does not need to convergent, by definition of a formal power series. – Qmechanic Feb 09 '23 at 12:27
  • From the first line of the Wikipedia page "Formal power series": "A formal series is an infinite sum that is considered indipendently of..."

    Also throughout the article they seem to consider only infinite sums. Do you maybe have another reference where they talk about formal power series with a finite number of terms to represent functions?

     – Siupa Feb 10 '23 at 09:31
  • I updated the answer. – Qmechanic Feb 10 '23 at 10:05
  • Thank you, I think I understand better now – Siupa Feb 10 '23 at 10:55