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This question is related to these others: mostly this one, but also this one and this one.

Do we care about CFTs in particle physics? Let me explain. Suppose we don’t know anything about string theory, holography, etc. And suppose we don’t know about the Wick rotation, so that we are unable to relate QFTs and statistical field theories. In this hypothetical world the only application of QFT is the Standard Model and its different extensions or possible minor changes. Would we be interested in CFTs? Why?

I'm asking this because I want to motivate the study of CFTs from the point of view of standard QFTs. The best argument I've found is that all "reasonable" QFTs are actually points in the RG flow from a fixed point at high energies $\text{CFT}_\text{UV}$ (UV completion) and a fixed point at low energies $\text{CFT}_\text{IR}$ (see for instance these lectures by Simmons-Duffin). Is the Standard Model an example of this? Looking at the running couplings of the SM, my answer would be no. Look for instance at this article by Rychkov. There he says there are basically three types of IR phases:

  • A CFT.

  • A theory with a mass gap. The example is QCD, because the beta function is negative and the coupling blows at low energies.

  • A theory with massless particles. The example is QED, because it is free in the IR, and at energies $E \ll m_e$ you're only left with photons. Someone said in the comments that this is a CFT (an empty CFT, with just the vacuum state for electrons) but Rychkov doesn't consider it as such.

So at least QCD is a counterexample of the above notion: it has no CFT$_\text{IR}$. On the other hand, for the UV part QED gives a counterexample: its coupling constant grows towards the UV, so it has no $\text{CFT}_\text{UV}$. Is this reasoning correct? If it is, then why are the only real-world QFTs (QED and QCD) a counterexample of such an extended notion (that all QFTs are points in the RG flow between two fixed points)? The "good" (conformal) behaviour of QCD at high energies (and arguably QED at low energies) seems accidental, not fundamental. So why do we care about CFTs in particle physics?

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  • $\begingroup$ "...why are the only QFTs that work (experimentally, if you wish) a counterexample of such an extended notion?" That question can be turned around: Why would we expect those QFTs that are most directly relevant to the real world to have UV completions that are also QFTs (CFTs), if we don't expect the most fundamental theory of the real world to be a QFT? $\endgroup$
    – Chiral Anomaly
    Commented Aug 15, 2019 at 2:00
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    $\begingroup$ Any QFT flows to a CFT in the IR. QCD is not a counterexample, but in that case the IR theory is trivial: it is an empty theory (or a free theory of Goldstone bosons if you forget about the quark masses). $\endgroup$
    – M.Jo
    Commented Aug 15, 2019 at 13:05
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    $\begingroup$ @MBolin The "theory with a mass gap" is the QFT. But in the deep IR, below the mass gap, there is only an empty theory (which happens to be a CFT): if you don't have enough energy to create at least one particle, the only state that exists is the vacuum. I think all of what I'm saying is pretty standard, but often people use the word "CFT" to talk about theories that are neither free nor empty. This might be why Rychkov makes a distinction between a flow to a non-trivial CFT and a flow to a theory with a mass gap ($\equiv$ an empty CFT). $\endgroup$
    – M.Jo
    Commented Aug 15, 2019 at 15:28
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    $\begingroup$ @MBolin My first comment only meant that I don't know of any compelling argument that every perturbatively-defined QFT should have a UV completion which is also a QFT. I don't know if the Standard Model has such a completion or not. I don't even know of any non-perturbative construction of the Standard Model on a lattice. $\endgroup$
    – Chiral Anomaly
    Commented Aug 16, 2019 at 0:05
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    $\begingroup$ Some points: 1) QED is simply not UV-complete, at least I don't know any UV-completions. Same for Standard Model, but in principle you can run its coupling constants to extremely high energies (even up to Planck scale maybe? -- don't remember) before encountering any problems of this sort. 2) Even if you don't know about Wick-rotation, Lorentzian CFTs still describe long-distance physics of quantum critical points. ... $\endgroup$
    – Peter Kravchuk
    Commented Aug 24, 2019 at 2:26

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