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

lonelyphysicist

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

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.

Last edited:
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