# Question about epsilons that occurs in calculating beta functions in QFTs w dim reg

by lonelyphysicist
Tags: beta, epsilons, functions, occurs, qfts
 P: 32 In deriving the beta function of, say, QED using dimensional regularization we get the relation (up to 1 loop) $$\beta[e] = - \frac{\epsilon}{2} e - e \frac{d ln[Z_{e}]}{d ln[\mu]} \quad (1)$$ and $$Z_{e} = 1 + \frac{e^{2} A}{\epsilon}$$ where e is the coupling, $Z_{e}$ is the renormalization of the coupling, $\mu$ is the arbitrary scale introduced to make the dimension of the coupling the same as if $\epsilon$ was zero, and A is some constant that does not depend on $\mu$ or $\epsilon$. Now if I do the following math $$\frac{d ln[Z_{e}]}{d ln[\mu]}$$ $$= \frac{1}{1 + \frac{e^{2} A}{\epsilon} } \frac{d}{d ln[\mu]} \left( 1 + \frac{e^{2} A}{\epsilon} \right)$$ $$= \frac{2 e A}{\epsilon + e^{2} A} \beta[e]$$ Is this correct? It seems extremely elementary, but if I trust my results $$\beta[e] \left(1 + \frac{2 e^2 A}{\epsilon + e^{2} A} \right) = - \frac{\epsilon}{2}e$$ Taking the limit $\epsilon$ going to zero I get that the beta function for QED is zero up to one loop! What seems to be usually done is we taylor expand $\frac{1}{1 + \frac{e^{2} A}{\epsilon} } \approx 1 - \frac{e^2 A}{\epsilon}$ and so $$\frac{d ln[Z_{e}]}{d ln[\mu]}$$ $$= \frac{2 A e \beta[e]}{\epsilon} + \mathcal{O}[e^{4}] \quad (2)$$ and next we put (2) into (1) and iterate $$\beta[e]$$ $$= - \frac{\epsilon}{2}e - \frac{2 A e^{2} \beta[e]}{\epsilon} + \mathcal{O}[e^{4}]$$ $$= - \frac{\epsilon}{2}e - 2 A e^{2} \left( - \frac{\epsilon}{2}e - \frac{2 A e^{2} \beta[e]}{\epsilon} + \mathcal{O}[e^{4}] \right) \frac{1}{\epsilon} + \mathcal{O}[e^{4}]$$ $$= - \frac{\epsilon}{2} e + A e^{3} + \frac{(2 A e^{2})^{2} \beta[e]}{\epsilon^{2}} + . . .$$ and we say as epsilon goes to zero it's approximately equal to $$A e^{3}$$ (For instance $A = 1/12 \pi^{3}$ for pure QED.) How can we taylor expand in powers of e when $\epsilon$ is supposed to be tiny in dimensional regularization? Even if we do taylor expand and iterate what about the $1/\epsilon^{2}$ and possibly even more infinite quantities? This issue really bothers me a lot because I'm sure there's something I'm not understanding here, since the QED coupling does indeed run as calculated, right? Any clarification would be deeply appreciated.

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