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}(i\gamma^\mu D_\mu-m_{f,0})\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+|\partial_\mu\phi|^2-m_{b,0}|\phi^2|\\ \quad &+\left( \frac{\lambda_3}{2}|\phi|^2\phi+\text{h.c.} \right)-\frac{\lambda_4}{4}|\phi|^4-(y\phi\bar{\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?
