Is the hierarchy problem definitely a "problem"?

Question

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.

Answer 1

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. Note that in an alternative renormalization procedure, you can even skip explicit regularization altogether.
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 $n>0$. The order of magnitude estimation using $(\frac{Q}{M})^n$ is usually a good way to tell whether a BSM term is relevant (see an example here).

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 the presumably large ${M}$, 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 $M$.