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Do we really understand why the renormalization group in $d=2+\varepsilon$ and $d=4-\varepsilon$ taking $\varepsilon=1$ gives "good" values for critical exponents in $d=3$? Are they exceptions?
Is it also the case in high energy physics (particle, string, quantum gravity)?

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I would say we do for $4-\epsilon$, but $2+\epsilon$ is something I don't know. Where do you start for calculating Ising critical exponents in 2d? The 2d fixed points are special, and their perturbation theory doesn't seem to me to be an easy link to 3d. What's a reference for $2+\epsilon$? –  Ron Maimon Apr 30 '12 at 4:48
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S. Ginsburg (or Ginzburg. I do not know exactly, how his name has been translated) has published a paper on a true 3D renormgroup in Sov. Phys. JETP in 1975, as much as I remember. In his approach all these artificial tricks with small ε that later appears to be equal to 1 are not present, and a rigorous renormgroup theory has been built. The results, say for the position of the fixed point was, however, the same as that in the Wilson's theory. My understanding is that one can use the ε-expansion just as a trick that leads to a correct answer (just as it is established empirically), but those who do not like it and want to be safe should go for the 3D renormgroup of Ginsburg. It is, by the way, not more complicated than the ε-expansion approach.

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