Renormalization Group and Dimensional Regularization

Currently I am learning about regularization, renormalization and the renormalization group. In particular, a lot of detail is devoted to dimensional regularization. There are a couple of things I struggle with when discussing the renormalization group, particularly in the context of DimReg. The notes are part of a book that is about to be published, so they are not available online. I will state some of the comments/outlines that are made, after which I will explain the things that are unclear to me.

1. Renormalization of QED

After computing the typical divergent Feynman integral in $4 + epsilon$ dimensions and extracting the $epsilon$-poles it is discussed how to cancel these infinities. Counterterms are introduced. The only thing the counterterms are required to do is cancel the infinities generated by loop diagrams. So, there is an ambiguity in defining the counterterms as we are free to absorb finite terms into the counterterms. In particular, two schemes are discussed. In the MS scheme, where we only include the pole, the counterterms take the form $$Delta mathcal{L} propto frac{1}{epsilon}.$$

In the $overline{mbox{MS}}$ scheme we include the pole together with some finite terms into the counterterms such that $$Deltamathcal{L} propto left[frac{1}{epsilon} + frac{1}{2}gamma_E – frac{1}{2}ln4piright].$$

Having computed the counterterms, we try to absorb the infinities into the bare parameters of the theory. In the $overline{mbox{MS}}$ scheme we obtain for example $e_0 = mu^{-epsilon/2}Z_e$ with $$Z_e = 1 – frac{e^2}{12pi^2}left[frac{1}{epsilon} + frac{1}{2}gamma_E – frac{1}{2}ln4piright],$$

where $e_0$ is the original, or bare charge (which is infinite), and $e$ is the renormalized charge. If we were to change the scheme, the definition of the renormalized charge would change such that $e_0$ would stay the same. $mu$ is the auxiliary mass parameter keeping track of mass dimensions. Now it is stated:

After renormalization the parameter $mu$ will characterize how the finite terms have been separated from the infinite ones.

2. Renormalization Group

In the intro the following is stated:

The renormalization procedure is not unique since counterterms are only defined up to finite terms. We are therefore dealing with a continuous variety of renormalized theories representing the same unrenormalized theory. When changing the renormalization prescription, the finite quantities we are interested in change. The renormalized parameters will also change, such that the bare parameters will stay the same […]. We will discuss the renormalization group equations in the context of dimensional regularization. The variety of prescriptions can be parametrized by the auxiliary mass parameter $mu$

Consequently, we discuss the RG equations in the context of the MS-scheme. We then have an expansion of unrenormalized coupling constants $g_{0alpha}$ of the form $$g_{0 alpha} = mu^{d_alpha epsilon}left[g_{alpha} + sum_{i = 1}^{infty} frac{a^i_{alpha(g)}}{epsilon^i}right] (*).$$

It then states:

The subtractions defined by (*) are not unique, and it is always possible to introduce additional finite terms. Here only poles are retained such that we have a one-parameter set of renormalization conditions characterized by a single mass parameter $mu$. […] Changing the value of $mu$ corresponds to adopting a different coupling constant g, in accordance with (*)

3. My problems

From the section on the Renormalization of QED I got the feeling that the mass parameter $mu$ defines how we separate finite from infinite terms. Or, in other words, how we define our renormalized parameters. This would suggest to me that very bluntly, one value of $mu$ would correspond to MS, and another value of $mu$ would correspond to $overline{mbox{MS}}$. But this does not make sense to me, as for example in equation (*) it states that we retain only poles, so we use the MS scheme. But the $mu$ is still there! If we vary $mu$, the renormalized coupling $g$ changes, but we still only retain poles. So no matter what we pick for $mu$, we will still be doing minimal subtraction (MS). This is also what I understand from the claim of the authors when they say "a one parameter set of renormalization conditions". This seems that the ‘non-uniqueness of (*) the authors refer to is only determined by picking different values of $mu$, and not absorbing different finite terms.

But then it seems to me that there are 2 dimensions of arbitrariness. One is how many finite terms we choose to include in the counterterms, and the second one is the value for $mu$ we pick. They seem completely unrelated. So how can, as the authors claim, $mu$ parametrize the way in which finite is separated from infinite?

Furthermore, I know that when doing calculations we at one point want to pick $mu$ "close to the energy scale of the process we are considering". But what does this even mean, ‘picking this $mu$? In the interpretation of the $mu$ as parametrizing separate from infinite, this would suggest that the energy scale of the process we consider tells us how many/what finite terms we need to include in the counterterms. But we compute things after we have made this decision (so, for example after we chose the MS scheme). So how can this be?