Doubts in understanding the role if quantum corrections in the Hierarchy Problem Trying to understand the Hierarchy problem many questions come to my mind that I am unable to answer due probably to my poor understanding of renormalization.
The basic set up of the hierarchy problem, the way I currently understand it, is the following. Because of the so called quantum corrections the mass of the Higgs mass gets contributions that are of the order of the UV cut-off square, giving astronomical corrections to the Higgs mass.
If I am not mistaken, these quantum corrections come from the fact that in the full Higgs propagator there is a $\Xi$ self-energy term alongside the mass term
$$\frac{1}{p^2-m^2-\Xi}$$
here I am working with what Peskin calls renormalized perturbation theory, where we are considering the counter terms in the Lagrangian and $m$ is already the renormalized Higgs boson mass.
Here already a question arises. I have often read that $m_0$, the bare mass is not the real mass, but rather that we have to consider the renormalized mass as the try physical mass which is measured. Nonetheless, the way in which the Hierarchy problem is presented suggests me that for the true mass of a particle, even after renormalization, we have to consider the self energy. So here goes my first question: Which is the measured Higgs boson mass, $m$ the renormalized bare mass, or $\sqrt{m^2-\Xi}$? In case that the answer is the latter, the why do we keep callin $m$ the physical mass?
I have stated that the quantum corrections to the Higgs mass have to be of the order of the UV cut-off square. Nonetheless, then I read (in case you wonder I am reading the intro of this review and https://www.weizmann.ac.il/particle/nir/uploads/file/chapter1.pdf) that if we regulated the loop integrals with ,say, dimensional regularization the UV cut-off doesn't show up but we have anyhow a problem if because the next term in the quantum corrections is proportional to the square of the fermion in the loop, which in case it were big would give big corrections.
So, if I understand correctly, the Hierarchy problem ,in some sense, is different depending on how you regulate your loops,because the reason for the problem is different in each case. Second question, how can this be so? I mean, if there are no heavy particles in the loop we would have no problem with dimensional regularization, but even in that case we would get huge corrections with a cut-off. 
Also, the actual correction to the Higgs mass is different in both cases, so then the full Higgs mass is different in cut-off and dimensional regularization. My third question is then, which is the real Higgs mass?
And as a last question, in dimensional regularization the quantum corrections depend on an arbitrary scale $\mu$, but not in the cut-off regularization case. Then we see that the corrections are a function of some variable in one case, but not in the other, how can this be so?
 A: For your first question: the physical mass is the measured mass. If $m$ is not the measured mass, don't call it the physical mass, because it isn't. If someone does, correct them. :-)
For your second question, I think we should look at the physical interpretation of all those mathematical manipulations. The hierarchy problem (as I understand it) comes from the fact that the Standard Model (and the Higgs) are low-energy approximation to some Theory of Everything. That means that, in that Theory of Everything, there are heavy particles in the loops. You can try to include those heavy particles (add some at random and see what they do) and use dimensional regularization; or you can just say the SM is valid up to a certain scale, slightly below the masses of those heavy particle, and put a cut-off there. There results in both cases will be similar. That "common lore" that dimensional and cut-off regularization give the same results is only valid if UV physics don't come into play, which is not the case in the hierarchy problem.
(I think that also kind-of answers your third question.)
For the last question: Quantum corrections depend not only on that scale $\mu$, but also on counterterms. Those counterterms are also arbitrary up to a certain point, such that the arbitrariness will also creep into cut-off computations (but in a slightly sneakier way). 
