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I am trying to understand renormalization in Wilson approach.

There's cool picture, which demonstrates flow of theories in IR:

So, if one interested in UV limit, one need reverse flow and flow in this reverse direction.

As it clear from picture, this procedure can be done only for red line. Dashed or solid line will lead us in UV to infinite values of couplings.

But in 6 Lectures on QFT, RG and SUSY there's the picture, which illustrate continuous limit:

As I understand, the idea is in construction on some lines of theories, that will lead to finite couplings values on infinite energy scale.

But this looks incorrect, because in finite scale $\mu$ these theories have different IR limits.

So, I don't understand, how to take continuous limit, if effective theory doesn't lie on renormalized trajectory?

Nikita
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  • Please do not post images of texts you want to quote, but type it out instead so it is readable for all users and so that it can be indexed by search engines. For formulae, use MathJax instead. – ACuriousMind Aug 05 '20 at 02:00
  • Note also that you have omitted a crucial part of the reference that you are quoting - it is specifically about conformal field theories, which have a feature called "universality", and it explicitly says that it is universality that allows the procedure outlined in your quote to work. Can you perhaps ask a more specific question what you don't understand about this? – ACuriousMind Aug 05 '20 at 02:03
  • I don't quite understand your question. However, I can say that IR limits (with finite $\mu$, i.e. you didn't flow all the way to the IR) being "different" is usually not a problem: the different IR actions will differ from each other by some RG irrelevant terms which we do not care about. As you truly take $\mu \to 0$, the IR limit becomes unique as the irrelevant terms all disappear from the action. – physics Aug 05 '20 at 02:39
  • @physics, I edited my question. But in every fixed scale $\mu$ such theories will be different. I don't understand, how to save this property in continuous limit. – Nikita Aug 05 '20 at 07:57
  • @ACuriousMind, I edited question. I think that it is better to paste such picture to save wholeness of author statements. – Nikita Aug 05 '20 at 08:00
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    As I said, pictures of text are suboptimal because they cannot be indexed by search engines and are unreadable for users relying on screen readers or similar technology. There is nothing lost if you type the text instead. I also don't think you have really made your question more specific and you still haven't included the relevant context about CFTs and universality in your question. Answerers should not have to click through to your sources to understand what you're actually asking about. – ACuriousMind Aug 05 '20 at 11:23
  • @ACuriousMind, I am familiar with CFT context and I understand relation to CFT. I can not understant renormalization of theories outside of renormalized trajectory.So I focus on this. Also, I don't understand, why you always mention CFT, my question about other aspect. Maybe you meen UV fixed poin, but as I undersytand, It is possible to take take UV limit without UV CFT. I wanna clarify this. – Nikita Aug 05 '20 at 11:53
  • @physics: Saying the irrelevant couplings disappear is a common mistake in the present context. If you do statistical mechanics, i.e., you have a fixed lattice theory and you want to know how does it look at large distance using the RG method, then what you said is correct. But this is not the situation discussed here at all. In fact, simply because the RG is nonlinear, the final QFT or the curve "renormalized trajectory" will be made of points whose (linear) coordinates (i.e., coefficients of individual terms in the action) have nonzero irrelevant components. It is possible... – Abdelmalek Abdesselam Aug 05 '20 at 15:01
  • ...for you to be right nevertheless, but for this one would need a switch to curvilinear coordinates adapted to the RG flow (Wegner's concept of nonlinear scaling field). – Abdelmalek Abdesselam Aug 05 '20 at 15:02

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I explained all this in I hope sufficient detail at

What is the Wilsonian definition of renormalizability?

and I urge the OP to read it. However, let me give some quick remarks here.

There is not much to say about the first picture which just shows how the RG flow looks like near the UV fixed point that one is trying to perturb in order to construct a new family of QFTs in the continuum. Except one could say that in order to properly interpret the picture, one has to keep in mind that the coordinates in the picture are dimensionless couplings and not dimensionful ones typically used in HEP. The second picture deserves more comments as it provides a pedagogical cartoon for how to take a continuum limit. In "continuum limit" the important word here is "limit". When the dust settles, one has removed the cutoff etc., the final product or continuum QFT is the curve labelled "renormalized trajectory". It is obtained as a limit of other curves. For each $\mu'$, you pick a starting point $\tilde{g}_i(\mu')$ and you run the RG from there. That gives you a curve. If the starting points are well chosen, these curves will converge to the limiting curve "renormalized trajectory".

Note that the picture illustrates a very general approach (standard plus more exotic constructions). The simplest and most common way (standard construction) to pick the starting points is as points on the tangent at the fixed point to the "renormalized trajectory". Namely, as $\mu'\rightarrow \infty$, the initial values $\tilde{g}_i(\mu')$ will converge to the fixed point itself. Think, for example, of a Gaussian fixed point, and the $\tilde{g}_i(\mu')$ only containing relevant terms (in fact only one if the 1st picture is to be trusted as saying there is only one relevant direction). The more exotic construction suggested by the 2nd picture is when the $\tilde{g}_i(\mu')$ converge not to the fixed point but to a point on the stable manifold of that fixed point. If you want an example where this "exotic" construction done in a mathematically rigorous and nonperturbative way, see Theorems 5 and 6 in Section 9 of the article

https://arxiv.org/abs/1302.5971

Abdelmalek Abdesselam
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