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(I have already read this post but my question is different)

Reading Ch. 12 of Weinberg's Quantum Field Theory Vol. 1, he states that all realistic (interacting) QFTs are now believed to be EFT of something else (string/M theory I assume).

Also in the Zinn-Justin book QFT and Critical Phenomena in the intro to Ch. 8 and 9 he says the same thing, seeming to imply that no (interacting) LOCAL QFT could be UV-complete.

I know that the $SU(3)×SU(2)×U(1)$ theory is not UV-complete, so we can think of it as an EFT.

  1. First point:

Suppose we find a gauge group $G$ (a single one like $SU(N)$ or a product of multiple groups) with an associated gauge theory, not necessary a Yang-Mills one for the gravity sector and that comprise all the relevant fermion/scalar reps such that the resulting $SU(3)×SU(2)×U(1)×G$ is UV-complete and contains all the physics up to now.

Is there a reason to consider such a theory an effective one? I know it's not a unified theory, but is this a reason to consider it an EFT or it's just a decomposition of such final unified theory?

  1. Second Point:

Forgetting the standard model theory and the above example, IF an interacting local QFT can be UV-complete, ie if such a local and UV-complete theory exist, is there a reason to consider it an EFT of something deeper?

Is there a reason why we should consider as fundamental something different from "standard" QFTs?

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  • $\begingroup$ Could you please state exactly what part of Chapter 12 in Weinberg you are talking about? I cannot tell what you are referring to $\endgroup$ Commented Jun 29, 2023 at 19:01
  • $\begingroup$ And is this the passage from Zinn-Justin? books.google.ca/… $\endgroup$ Commented Jun 29, 2023 at 19:05
  • $\begingroup$ I can't see the passage from Zinn-Justin, by the way it's the first page of chapter 8 and for Weinberg the first page of chapter 12 $\endgroup$
    – LolloBoldo
    Commented Jun 30, 2023 at 8:28

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  1. No theory of the form $U(1)\times G$ can be UV complete, since the $U(1)$ factor has a Landau pole (in 4d). UV complete gauge theories always involve semi-simple groups (although this is not enough: the number of fermions cannot be too large, otherwise the beta function changes sign).

  2. If a theory is UV complete, we do not think of it as an effective theory, but you definitely could, if you want to. For example, semi-simple theories as in the above point, when UV complete, can still be thought of as an effective theory of a more fundamental theory, e.g. something involving gravity.

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  • $\begingroup$ BTW, I'm not entirely sure what you mean by "IF an interacting QFT can be UV-complete". A theory either is, or it isn't, so "can" makes no sense here. Do you perhaps mean "if a theory can be UV-completed"? $\endgroup$ Commented Jun 23, 2023 at 20:27
  • $\begingroup$ Also, i never stumbled upon semi-simple theories, do you have any reference to take a quick look at them? $\endgroup$
    – LolloBoldo
    Commented Jun 23, 2023 at 20:48
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    $\begingroup$ The reason i used "can" is because Zinn-Justin says that no matter what the theory is, local interacting QFT are never UV-complete. So the "can" had the meaning of "if such a theory could exist in the first place". I edited the original text to make it more clear $\endgroup$
    – LolloBoldo
    Commented Jun 23, 2023 at 20:54
  • $\begingroup$ @LolloBoldo semi-simple just means direct sum of simple algebras, it is just a fancy way of saying: the gauge group is a product of $SU(N)$ factors (there's also the orthosymplectic but nevermind). In other words, semi-simple just means that there are no $U(1)$ factors in your gauge group. $\endgroup$ Commented Jun 24, 2023 at 22:18
  • $\begingroup$ @LolloBoldo Haven't read Zinn-Justin in a very long time so not sure what exactly he means by that, but there are lots of UV complete, local, interacting QFTs. Semi-simple theories are examples in 4d, and in lower dimensions almost everything is UV complete. In d>4 it is very hard to find non-trivial UV complete theories, but in $d\le 4$ there are infinitely many examples... $\endgroup$ Commented Jun 24, 2023 at 22:19
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By my reading, Zinn-Justin is more emphatic than Weinberg. Zinn-Justin seems to think that the very presence of ultraviolet divergences in a local QFT means that it is necessarily incomplete; whereas Weinberg merely says that the theories used today are "generally believed" to be incomplete, and doesn't state the reason, so he could be talking about quite different issues like gravity, grand unification, or the hierarchy problem.

As @AccidentalFourierTransform writes, a theory with a U(1) factor in its gauge group will have a Landau pole, but a theory with a semi-simple gauge group doesn't have this problem, and may be regarded as UV-complete. There is a little debate about this here, using the example of QCD.

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Just to give a slightly different perspective than the other answers (which I do agree with).

There are two questions, a mathematical and a physical one. The mathematical question is whether a given quantum field theory can be defined on a lattice, and we can take the lattice spacing to zero, in such a way that the field theory is well defined in the limit. I believe there is good evidence that some theories are UV complete in this sense, such as conformal field theories and maybe QCD.

The physical question is whether the quantum field theory that we currently use to describe our Universe -- the Standard Model -- is UV complete. Or, more generally, if we will be able to come up with a UV complete quantum field theory that is capable of completely describing our reality. The obvious problem with the Standard Model is that it does not contain gravity, so we're really interested in the more general version of the question. The program of asymptotic safety in quantum gravity proposes that it is possible to model gravity and all matter we've observed as a UV complete quantum field theory. However, no one knows if that approach works. Other approaches to quantum gravity, such as string theory, replace quantum field theory with something else, in which case the most complete model we could use to describe our Universe would not be a quantum field theory.

In some sense, the point of effective field theory is that we don't need to know whether what theory represents physics at energies beyond what we can probe; Weinberg argues that any theory will look like a quantum field theory in the low energy limit, whether the UV complete theory actually is another quantum field theory or if it's something different like string theory.

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