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My question is about figure3 (page 8) of this paper hep-th/9705131. Start from Seiberg-Witten theory, integrate out the charged high energy modes down to Higgs scale and we get a $U(1)$ gauge theory without matter. Then the author claims that the beta function is zero.

Is there any argument or calculation for this?

It seems contradict with formula (2.7) Arkani-Hamed and Murayama (hep-th/9707133). $$\frac{1}{g^2_{eff}}= \frac{1}{g^2_{h}}+\frac{1}{4\pi^2}\log{\frac{v^2}{M^2}}+\rm{non-pertubative} $$

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The claim is consistent with page 37 of this and page 11 of this. The $\beta$-function of $U(1)$ depends on the charge of the degrees of freedom, but since there are none below the scale $a$, it is zero. As for hep-th/9707133, I suppose that the coupling there is given for non-vanishing charges.

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