Source of hierarchy problem for fermions and bosons

Question

In the beginning of a SUSY course, we computed $1$-loop level corrections to the mass of a bosons $phi$ and a fermion $psi$ in the theory
begin{align}
mathcal{L} &= bar{psi}(igamma^mu D_mu-m_{f,0})psi-frac{1}{4}F_{munu}F^{munu}+|partial_muphi|^2-m_{b,0}|phi^2|
quad &+left( frac{lambda_3}{2}|phi|^2phi+text{h.c.} right)-frac{lambda_4}{4}|phi|^4-(yphibar{psi}psi+text{h.c.})
end{align}
i.e. the theory is QED with a scalar, a cubic interaction, a quartic interaction and a Yukawa interaction. The corrections $delta m_{f,0}$ and $delta m^2_{b,0}$ such that
begin{align}
m_{f} &= m_{f,0}+delta m_{f,0}
m^2_{b} &= m^2_{b,0}+delta m^2_{b,0}
end{align}
at $1$-loop that we obtained had different dependence on $Lambda$, the cut-off energy scale:

$delta m_{f,0}$ is proportional to $m_{f,0}$ so the corrections are still small if $m_{f,0}$ is small
$delta m^2_{b,0}$ can be arbitrarily big, as $Lambda$ gets big, regardless of $m^2_{b,0}$

we then say that the mass of $psi$ is protected or technically small, so there is no hierarchy problem for fermions, and that the mass of $phi$ is not protected so there is a hierarchy problem for bosons.

What is the fundamental reason behind the fact that the mass of fermions is protected while the one of bosons is not? In other words, are the long computations of the loop diagrams the only reason or is there something more fundamental?

Is this true in general or just in this theory? If it's true in general, why is that? (related to question 1).

What does we really mean by "there is (or not) a hierarchy problem" in this case?

Chiral Anomaly · Answer

If the fermion didn't have mass term, then the model would have more symmetry (chiral symmetry). Actually that's not quite necessarily true, because the chiral anomaly can ruin chiral symmetry, but the chiral anomaly is really an infrared (low energy) thing,$^dagger$ whereas the relationship between the bare and renormalizes masses is an ultraviolet cutoff-sensitive thing. So the argument still works: if the fermion didn't have a mass term, then the model might as well have more symmetry (chiral symmetry), so renormalization won't generate a mass term if the bare mass is zero. Heuristically, this suggests that renormalization won't change the bare mass much even if the bare mass is nonzero. [Note added two months later: Clearly this argument is far from rigorous, and https://arxiv.org/abs/0901.0150 says it's flat-out wrong. From the summary on page 17: In the one-flavor version of QCD, "all chiral symmetry is lost [because of the anomaly]. One consequence is that the mass of a non-degenerate light quark is unprotected from additive and scheme dependent renormalization.]
In contrast, setting the mass of the scalar field to zero doesn't increase the model's symmetry at all. Even if we start with a zero bare mass for the scalar field, renormalization will generate a nonzero renormalized mass, so the scalar mass is not protected: the relationship between the bare and renormalized masses depends strongly on the ultraviolet cutoff, and fine-tuning is needed in order to keep it far below the scale of the cutoff. And by the way, we typically need to make the bare mass-term negative (opposite of the self-coupling term) in order to keep the renormalized mass far below the cutoff. We can see this intuitively by thinking in terms of correlation lengths instead of masses, as I explain in my answer to another question.

$^dagger$  Chiral anomalies are often blamed on the need to use a regulator, which can make them seem like an ultraviolet thing, but that's really not the best perspective. The purpose of using a regulator is simply to define the theory. Only after it's defined can we ask well-posed questions about it, like whether or not it has an 't Hooft anomaly (chiral anomalies are a special case of this), and then we can see that chiral anomalies are really an infrared thing, indepenent of the details of the regulator. Contrast this with the relationship between bare and renormalizes masses, which does depend on the details of the regulator (such as the precise value of the ultraviolet cutoff).

Source of hierarchy problem for fermions and bosons

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