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There have been a bunch of questions related to the hierarchy problem, but I still can't help but feel that an assumption is being made that is not backed by any example of correctness and is thus potentially unwarranted. Of all of the questions already posted, I would say mine is most closely related to this one.

My understanding of the hierarchy problem is the following: in the context of the standard model thought of as an effective field theory with some cut-off $\Lambda$ (typically thought of as being far above the electroweak scale), the bare Higgs mass must be incredibly finely tuned so that it cancels its quantum corrections, quadratic in $\Lambda$, to leave behind the relatively small physical Higgs mass.

My issue with the supposed "problem" is this: why is the view that the standard model is an effective field theory with a cut-off any more valid than the view that it is a renormalisable theory where the cut-off is just a part of the regularisation/renormalisation scheme, eventually taken to be arbitrarily large (i.e. infinite)? In the latter picture, any contribution which is divergent with increasing $\Lambda$ is infinite as $\Lambda$ goes to infinity, so I don't see how a quadratic divergence is actually any worse than a logarithmic one.

The Higgs mass is the only parameter in the standard model that is dimensionful, so there is no other example to demonstrate that the idea that physical, dimensionful quantities should have values on the order of a suitable power of the cut-off is correct. In fact, outside of the standard model, the cosmological constant has the same problem; in these two cases of reasoning for the values of dimensionful parameters, the naturalness argument seems to fail terribly both times.

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  • $\begingroup$ I don't have time for a full answer, but one source is Cliff Burgess' notes "Introduction to Effective Field Theory": arxiv.org/abs/hep-th/0701053, see the discussion after Eqs 31 and 50. In particular, you are right that cancelling the cutoff dependence is not the real problem; the problem is that there are finite "threshold corrections" involving large mass scales that arise when matching the low energy EFT to the more complete high energy theory. Having said this... at the moment proposed solutions to the hierarchy problem have not had much success. $\endgroup$ – Andrew Feb 26 at 20:20
  • $\begingroup$ Thanks Andrew, I'll take a read tonight! $\endgroup$ – turbodiesel4598 Feb 26 at 20:50
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The hierarchy problem has to be framed in the context of beyond standard model physics. You have to distinguish between 5 mass scales, namely

  1. $m$: the mass of the particle in concern, e.g. Higgs mass $m_H$.
  2. $\Lambda$: the UV cutoff scale of the regularization scheme (in dimensional regularization (DR), $\frac{1}{\epsilon}$ plays the role of $\Lambda$, where $\epsilon = d -4$). At the end of the renormalization procedure, $\Lambda$ can be safely sent to infinity (or $\epsilon$ sent to zero in DR), thanks to the painstakingly crafted counter terms.
  3. $Q$: the energy scale of the incoming/outgoing particles involved in a scattering process.
  4. $\mu$: the renormalization scale, which is an arbitrary scale to anchor the scattering amplitude (or coupling 'constant') as a function of $\frac{Q}{\mu}$ (or $ln(\frac{Q}{\mu}$)). The renormalization scale $\mu$ is a fiat scale that is set forth by human convention/convenience. Usually $\mu$ is set to the typical energy scale $Q_0$ of a scattering process. See more explanations about the renormalization scale $\mu$ here.
  5. $M$:the mass scale where beyond standard model (BSM) physics effect comes into the picture. $M$ could be either the grand unification scale $M_{GUT}$ or Planck scale $M_P$. In the effective field theory framework, the BSM Langrangian terms are suppressed by a factor of $(\frac{Q}{M})^n$, with $n>0$.

Assuming that there are BSM Langrangian terms, the hierarchy problem concerns the uncanny fine-tuning to arrive at the tiny value of ${m}$ compared with ${M}$, unless there is a spontaneously broken symmetry (technical naturalness) constraining the otherwise large BSM quantum loop corrections (of order $M$) to $m$.

As you can see, the hierarchy problem has to do with BSM $M$, but not cutoff $\Lambda$. If there is no $M$, the quadratic divergent corrections to Higgs bare mass is of the order $O(\Lambda^2)$, which can be canceled out by the $\Lambda$-dependent mass counter term. And the cutoff $\Lambda$ can be safely sent to infinity without any issue. Thus there is no hierarchy problem if there is no $M$.

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  • $\begingroup$ Thanks very much - I think a couple of years ago I read about this, i.e. the difference between $\Lambda$ and $M$, but forgot, because if I'm not mistaken, conversations about effective field theory seem to often treat $\Lambda$ and $M$ as the same thing, or at least seem to. Although I would say this answers my opening question (and also answers the question I linked to!), does the follow-up issue still not exist, i.e. that we don't actually have an example of any dimensionful parameter in fundamental physics being on the order of the suitable power of $M$? If so, why should we expect it? $\endgroup$ – turbodiesel4598 Feb 26 at 21:00
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    $\begingroup$ @turbodiesel4598, one clue of $M$ is the light neutrino mass being of the order of $m^2/M$ in the seesaw scenario, where $m$ is the electroweak/Higgs scale,and $M$ is estimated to be close to the GUT scale. $M$ is actually the Mojorana mass of the heavier right-handed neutrino. $\endgroup$ – MadMax Feb 26 at 21:12

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