# How SUSY solves the hierarchy problem?

I am struggling to understand the argument for why the introduction of a stop in SUSY can solve the hierarchy problem. The quadratic divergence from the top loop in the higgs mass calculation gives a contribution of $$\delta m_h^2 = - \frac{|y_f|^2}{16\pi^2} \left[2\Lambda^2 + \ldots \right]$$ If we introduce a scalar $$S$$, there should be two additional diagrams to consider: and The first diagram should give a contribution like

$$\delta m_h^2 = \frac{\lambda_S}{16\pi^2} \left[\Lambda^2 + \ldots \right]$$

and the second like

$$\delta m_h^2 = \frac{|\lambda_S|^2}{16\pi^2} \left[\Lambda^2 + \ldots \right]$$

(I am probably off by some constants or something). Now, this seems to me to only cancel out of $$y_f = \lambda_S = 1$$, which is close to 1 for the top but not exact. Isn't this just another unnatural fine tuning problem?

You are considering three loop diagrams with three couplings: $$g_1$$ for the Higgs-top Yukawa coupling, $$g_2$$ for the $$h h f \tilde{f}$$ coupling, and $$g_3$$ for the $$h f \tilde{f}$$ coupling. When you do the loop diagrams, the contribution is schematically $$-g_1^2 + g_2 + g_3^2$$.
You've pointed out that this is not zero for general values of $$g_1$$, $$g_2$$, and $$g_3$$, but of course it isn't! Without further structure, these couplings have nothing to do with each other. The point of supersymmetry is that it does relate the couplings -- and if you work through the calculations in detail, it relates $$g_1$$, $$g_2$$, and $$g_3$$ in exactly the way needed to make the $$O(\Lambda^2)$$ part of the loop integrals cancel out.