What Explains the Extra Factor of 2 in Peskin and Schroeder's Eq. 18.84?

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Homework Statement

So I am self-studying the book of Peskin&Schroeder, and there is something I don't understand on page 616.

In eq. 18.80, there is a numerical factor of ½ and going from e² to α will introduce a factor 4π when proceeding to eq. 18.84. But then there should be a numerical factor of 2π in eq 18.84. What gives an extra factor of 2?

Homework Equations

(this is the first time I post here, please do tell if uploading a page in this format is not acceptable)

The Attempt at a Solution

I believe I understand the rest of the calculation on this page.

 

[Dr.AbeNikIanEdL]

Well, equation (18.80) is for $e^+e^-\rightarrow e^+e^-$, equation (18.84) is for $e^+e^-\rightarrow \text{hadrons}$. As the text says, the two are related by the optical theorem.