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This question is related to this one, but the author of the most voted answer seems to no longer have an account here so I'll open a new question instead of commenting on their answer.

I originally thought that counter-term renormalization and Wilsonian renormalization have two different objectives. Counter-term renormalization varies the cutoff scale $\Lambda$ but not the renormalization scale $\mu$, whereas Wilsonian renormalization varies $\mu$ but not $\Lambda$. Thus any attempt to compare them is conflating $\mu$ and $\Lambda$. But the answer to the linked question says:

In counterterm renormalization, you are essentially studying Wilsonian renormalization, but with the intention of sending the cutoff to infinity at the end

which suggests there is a connection.

In Wilsonian renormalization we start with a theory far in the UV (possibly at the continuum) and integrate out high momentum modes until we reach our renormalization scale. The result will be an effective Lagrangian where parameter values may have changed and new terms may have been added/removed. In summary, $\mu$ goes from the UV towards the IR.

On the other hand, in counter-term renormalization (as in renormalized perturbation theory) we first fix $\Lambda$ and $\mu$ and find Lagrangian parameters so that predictable observables (such as the mass) agree with their experimental values at scale $\mu$. Part of this renormalization process will require defining counter-terms and finding their values. Once this is done we try to take a $\Lambda \rightarrow \infty $ limit. Here $\Lambda$ go from the IR to the UV.

In trying to relate the two I noticed that they appear to be inverses of one another except one has to do with $\Lambda$ and the other has to do with $\mu$: one starts with a continuum or small scale theory and eventually gets to the scale one is interested in, in the other one starts at the scale of interest and tries to recover the small scale or continuum theory. Are they in any way inverses of the same procedure? If so, I have two further questions.

First, the renormalization group flow is not always invertible which is due to the fact that we cannot recover lost information. However, if there is a connection between the two, it seems in counter-term renormalization we are doing exactly that, we are somehow getting information on small scales from large scale behavior. Isn't this a problem? Or does counter-term renormalization give you no information about scales other than $\mu$?

Second, while going along the the renormalization group flow one picks up and loses all kinds of interactions that were not present in the original Lagrangian. In counter-term renormalization such terms do not appear and only has the terms that one starts with. Is this because we change $\Lambda$ and not $\mu$?

Another perspective is provided by this answer which says

  • the renormalized Lagrangian is the Wilsonian effective Lagrangian
  • the bare Lagrangian is the fundamental Lagrangian
  • the counterterm is the term needed to compensate for the RG flow between them
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    $\begingroup$ General tip: Referring to an answer as "the most voted answer" is ambiguous as it may change in the future. $\endgroup$ Commented Jan 10 at 7:16
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    $\begingroup$ Chapter 1.2 of this dissertation discusses the relation between these, as does Section 1 of Georgi's review of EFT. $\endgroup$ Commented Jan 19 at 10:35
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    $\begingroup$ I think some of these issues are explained in my previous answer physics.stackexchange.com/questions/372306/… but with different notations. The RG map $RG[t_2,t_1]$ in that answer is the one taking as input the theory at scale $\Lambda=e^{-t_1}$, and producing as output the effective theory at scale $\mu=e^{-t_2}$. $\endgroup$ Commented Jan 19 at 20:11
  • $\begingroup$ @AbdelmalekAbdesselam Thank you. I learn something new every time I revisit that answer. $\endgroup$ Commented Jan 20 at 6:33

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Since renormalization by counter terms is often performed for renormalizable theories, we will mainly focus on such theories in the following.

Is it correct to say that these are inverse procedures of the same process?

In short, no. To interpret the usual renormalization by cutoff regularization in terms of Wilson’s method, one must first integrate out the high-energy modes to obtain the effective Lagrangian, and then take the continuum limit. The process of integrating out high-energy modes corresponds, in the language of cutoff regularization, to calculating the diagrams and determining the mass and coupling constants as functions of the cutoff. The procedure of taking the limit $\Lambda\to+\infty$ in cutoff regularization corresponds to taking the continuum limit in Wilson’s method. Whether starting from Wilson’s method and taking the continuum limit, or performing renormalization by cutoff regularization, these lead to equivalent results, and are not inverse procedures. In your question, the perspective of the continuum limit is overlooked. In both approaches, in order to obtain results in the continuum theory, taking the limit $\Lambda\to+\infty$ and fixing the parameters at the renormalization scale are necessary.

Of course, the continuum limit has good properties only for renormalizable theories, so in practical applications of effective theories, the cutoff is sometimes kept finite (such as in chiral perturbation theory, some BSM theories, etc.). The theory with a finite cutoff is difficult to interpret solely within the framework of standard perturbation theory and renormalization, so in this sense, Wilson’s effective action carries deeper physical meaning compared to standard perturbation theory. However, as long as one is considering a renormalizable theories, these approaches are equivalent (but… see also the Note below).

Does counterterm regularization provide no information about scales other than $\mu$?

In the case of a renormalizable theory, roughly speaking, this is correct. If the energy scale of interest can be described by some renormalizable effective action, parameters in this action, like coupling constants or masses, are insensitive to the physics at higher energy scales. To understand the physics at such higher energy scales, one must consider non-renormalizable interactions (i.e. irrelevant operators of renormalizable theory).

In counterterm regularization, no new terms appear, and only the terms initially present remain.

Assuming that the system is sufficiently close to the critical surface and that the action of the theory is in a form close to that of the theory on the critical surface, the same statement can be obtained in Wilson’s method. To understand this, one should focus on the fact that the theory obtained by taking the continuum limit lies on the critical surface of theory space, which usually can be described by a finite number of relevant parameters. In other words, although the process of integrating out the high-energy modes introduces a large number of new terms, when taking the continuum limit and moving the system onto the critical surface, many of these parameters can be expressed as functions of a finite set of relevant parameters. This is entirely consistent with the results obtained from standard perturbation theory (of course, this holds true only if one is considering a renormalizable theory).

[Note]: If theory is renormalizable, there is usually a finite number of relevant parameters. However it should be pointed out that even for renormalizable theories, it is sometimes not possible to take a meaningful continuum limit (such as $\phi^4$ theory). In this case, issues like Landau poles appear even in perturbation theory with cutoff regularization. According to Wilson’s prescription, a strict continuum theory cannot be obtained, so for theories like $\phi^4$ theory, we need to use a sufficiently large but finite cutoff instead of positive infinity, and the theory needs to be treated as effectively continuous, in strict sense. However, in practical calculations, we often use the limit $\Lambda\to+\infty$. This approach does not pose practical problems. For more details, please refer to appropriate textbooks on renormalization group theory.

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  • $\begingroup$ Thank you! As you pointed out, I don't think I appreciate the $\Lambda \rightarrow \infty$ limit and what it's doing. In fact, I recently asked a similar question another post. My understanding is that it's designed to get a "complete" theory, if possible. Is that really the only reason why it's so important? $\endgroup$ Commented Jan 20 at 6:29
  • $\begingroup$ Also, I believe I will have to read more on critical surfaces and the renormalization group, but you mention that near/on a critical surface the theory only has a finite number of parameters. This doesn't rule out new terms appearing in the Lagrangian, only that at most a finite number of them may appear. Is the fact that no new terms appear in, for example, $\phi^4$ or QED a coincidence? $\endgroup$ Commented Jan 20 at 6:31
  • $\begingroup$ “My understanding….” I cannot understand your statement. Please rephrase if necessary. “This doesn't rule out new terms…” New terms do appear, but the coefficients of them can be described by a finite number of parameters, and renormalization is not required. This corresponds to the fact that in standard perturbation theory, if the renormalization is performed for a very small number of parameters, the entire quantum effective action becomes finite. $\endgroup$ Commented Jan 20 at 13:03
  • $\begingroup$ Sorry, I had meant that we take the continuum limit because we seek a theory that works at all length scales. Of course it turns out that this isn't a reasonable demand to make for all theories (e.g. $\phi^4$), but my understanding is that this was the original motivation. Is there some other reason? $\endgroup$ Commented Jan 21 at 7:04
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    $\begingroup$ I used very bad terminology. I meant much larger “energy” scale. But I should call it as much “higher” energy scale. The reason we can take the cutoff to infinity is that high-energy physics does not affect low-energy physics. Every effective theory has own characteristic energy scale, and therefore, it cannot be applied to all energy (or length) scales. $\endgroup$ Commented Jan 25 at 6:04

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