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As far as I understand one classifies the renormalization group (RG) into the Wilsonian RG and the continuum RG. The Wilsonian RG gives finite predictions by introducing a cutoff $\Lambda$ and absorbs thereby the UV behaviour in the couplings. The flow then describes how the couplings have to change when we change the cutoff. In contrast to that the continuum RG makes things finite by grouping terms together according to a reference scale $\mu$. The flow describes again how the couplings have to change if we change $\mu$ and thereby describes which terms are contributing the most.

I have now 3 questions to the RG:

  1. Is this a correct way to think of these RG‘s?

  2. In which of these categories fits the functional renormalization group? Because it seems that the FRG with it‘s IR cutoff $k$ is intermediate.

  3. Why is the continuum RG called continuum? Is this because in Wilsonian RG one integrates out short distance modes and hence gets a theory on a “lattice“?

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    $\begingroup$ Short answer: the continuum RG (not standard terminology) is the restriction of the Wilsonian RG to the unstable manifold of a fixed point (renomalizable QFTs, or QFTs obtained by perturbations with relevant operators). Long answer: physics.stackexchange.com/q/372306 $\endgroup$ Commented May 8, 2023 at 16:33
  • $\begingroup$ Hi Silas: Since this is non-standard terminology, consider to include a reference that uses it. $\endgroup$
    – Qmechanic
    Commented May 9, 2023 at 8:58
  • $\begingroup$ @Qmechanic Schwartz uses this terminology in his book “Quantum Field Theory and the Standard Model“ (cf. chapter 23) $\endgroup$
    – Silas
    Commented May 9, 2023 at 14:19

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