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I'm studying 1 loop renormalization of QED using QFT by Ryder. On page 345,

$$e_B=e\mu ^ {\epsilon \over 2} \Bigg(1+{e^2 \over {12\pi ^2 \epsilon}}\Bigg).$$

differentiating the above equation gives, to order $e^3$

$$\mu {\partial e \over \partial \mu }=-{\epsilon \over 2}e+{e^3 \over 12\pi ^2}.$$

I don't get this equation What I get is

$${\partial e_B \over \partial \mu} ={e\mu^{{\epsilon \over2 }-1}\over 2}\Bigg(\epsilon +{e^2\over 12 \pi^2}\Bigg) +\mu^{\epsilon\over 2}{\partial e\over \partial \mu }\Bigg(1+{e^2 \over 4\pi^2\epsilon} \Bigg).$$

I don't find any connection between this two equations can any one please help, Your little time may save my lot of time.

ROBIN RAJ
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2 Answers2

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The elephant in the room is the following question:

How can we treat $\frac{e^2}{4\pi^2\epsilon}$ as subleading as compared to $1$? In dimensional regularization the parameter $\epsilon$ is supposed to be small.

The brief answer is that renormalization is first-and-foremost a perturbative formal power series in the coupling constant $e^2$. Secondly, each coefficient of this formal power series is a truncated Laurent series in $\epsilon$. Not the other way around.

Qmechanic
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As per @Qmechanic's suggestion, you should have gotten $$ 0= {\epsilon \over 2} e \Bigg(1+{e^2 \over {12\pi ^2 \epsilon}}\Bigg)+\mu {\partial e \over \partial \mu } \Bigg(1+{3e^2 \over {12\pi ^2 \epsilon}}\Bigg), $$ i.e. $$\mu {\partial e \over \partial \mu }=-{\epsilon \over 2}e \frac{1+{e^2 \over {12\pi ^2 \epsilon}}}{1+{3e^2 \over {12\pi ^2 \epsilon}}}\\ =-{\epsilon \over 2}e \Bigg(1+{e^2 \over {12\pi ^2 \epsilon}}-{3e^2 \over {12\pi ^2 \epsilon}} + O(e^4)\Bigg)= -{\epsilon \over 2}e+{e^3 \over 12\pi ^2}+ O(e^5).$$

You may then take $\epsilon \to 0$ "safely". The coupling then increases with energy.

Cosmas Zachos
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