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How do issues of naturalness arise when regularizing QFT using dimensional regularization? I can only recall ever seeing naturalness arguments (hierarchy problem, cosmological constant problem, etc.) phrased in terms regularizing with a cutoff, where naturalness issues arise when physical quantities are quadratically divergent in the cutoff scale.

Is it hard to see how the same naturalness issues are addressed using dimensional regularization? Are there some hidden assumptions involved in using dimensional regularization? Do you reach the same conclusions as you do using a cutoff, but only after also using the RG equations?

I recall being told that when dimensional regularization is used to remove power law divergences there is additionally some optimistic assumption being made about the UV physics, but I don't know if that's correct or relevant to this problem.

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up vote 6 down vote accepted

Check this review on the cosmological constant problem for a nice discussion of what hierarchies look like in different regularizations.

Here is the rough idea. In cutoff or Pauli-Villars regularization the counterterms are sensitive to the cutoff scale(s) $\Lambda$. But there is no such scale when using dim.reg. (only the renormalization point $\mu$, which is usually a low scale). So what do you get instead? You need to add new physics.

Consider adding a heavy particle to your theory with mass $M$. This contributes through loops to the counterterms for the cosmological constant ($\propto M^4$), masses ($\delta m^2\propto M^2$) and couplings ($\propto \ln(M/\mu)$). So when talking about, say, the mass term of the Higgs boson, we say it is quadratically sensitive to the scale of new physics.

Note that this is a completely physical statement - no unphysical regulator scale appears in the argument. This would be the same if people made the argument properly in cutoff regularization: adding new particles and then doing the renormalization to get rid of the regulator scale. Talking about the cutoff dependence of quantities is really a nasty, unphysical shortcut.

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I totally agree with you (as I expressed in this question physics.stackexchange.com/questions/32281/…). I guess that you agree that there is not naturalness problem related to the Higgs, at least for now. –  drake Jul 10 '13 at 4:14
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@drake Yes I think you're right. The naturalness problem with the Higgs is not a problem of the standard model per se, it is a problem with the SM embedded in a larger theory. In the notation of your question if $M\gg m_P$ then you have a fine tuning problem. I guess the answer to (first part of) your question is that people usually assume the existence of such heavy particles... (i.e., dogma) –  Michael Brown Jul 10 '13 at 5:09
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