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Some terms in $H_0$ can be absorbed into the definition of $\epsilon$. To understand the validity of this substitution, you must go back to chapter 7 and calculate according to eqn. (7.2)-(7.6). You will find "we can absorb the positive coefficient into $\epsilon$ to get..."


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Imagine a QFT with some particle content. Some of these fields will be massless and some massive. For simplicity, consider a massles scalar field and a massive scalar field with mass $M$. If we are working at some energy $E\ll M$, we won't see the massive field (as happened with the Higgs before LHC, for example). This is the IR CFT. Why IR? Because we ...


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I think that this nomenclature has nothing to do with the regulators. "UV CFT" and "IR CFT" actually refers to the end point of the RG flow which is triggered by a relevant operator that perturbs an UV fixed point, and it ends (barring certain non-unitary QFT) into a fixed point at lower energy scales, hence the name IR.



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