The fermion contribution to the Higgs mass is $$\Delta m^2_{H}=-\frac{|\lambda_{f}|^2}{8\pi^2}\Lambda^2_{UV}+\dots$$ And the scalar contribution is: $$\Delta m^2_{H}=\frac{\lambda_{s}}{16\pi^2}\Lambda^2_{UV}+\dots$$ Summing up:

$$\Delta m^2_{H}=\Lambda^2_{UV}\underbrace{\left(\frac{\lambda_{s}}{16\pi^2}-\frac{|\lambda_{f}|^2}{8\pi^2}\right)}_{\approx\ 0}+\dots$$

Thus, because the cutoff $\Lambda_{UV}$ is many orders of magnitude higher than the Higgs physical mass, the term in the paranthesis should be very small.

Now, in SM, this would seem like a miraculous cancelation. This is ugly, unnatural, and it looks fine-tuned. This begs for an explanation.

What Supersymmetry does is to predict that there is a symmetry between bosons and fermions, such that: $$Q|\text{fermion}\rangle = \text{boson}$$ $$Q|\text{boson}\rangle = \text{fermion}$$ the action of the operator $Q$ on a fermionic state transforms it into a boson state and vice-versa. Now, if there are two scalar bosons for each SM fermion (because fermions have two spin states), then the quantum corrections would neatly cancel, provided that: $$\lambda_{S}=|\lambda_{f}|^2$$

Is this the right explanation for how SUSY solves the hierarchy problem (at least the qualitative explanation), or am I missing something?



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