= Solution
The <hierarchy problem> begins with a large separation of scales: the weak scale is much smaller than possible unification or gravitational scales. A fundamental <scalar mass> is especially sensitive to a much heavier scale. In an effective description with ultraviolet cutoff $\Lambda$, one schematically finds
$$
m_H^2=m_{H,\mathrm{bare}}^2+\delta m_H^2,\qquad
\delta m_H^2\sim\frac{c}{16\pi^2}\Lambda^2.
$$
The coefficient $c$ contains gauge, scalar and <Yukawa couplings>, with different signs for bosonic and fermionic loops. The explicit quadratic cutoff dependence is regulator-dependent, but a heavy physical particle coupled to the Higgs produces a threshold correction of order its mass squared. That heavy-threshold sensitivity is the physical difficulty.
The <technical hierarchy problem> asks whether a small scale, once chosen, remains stable under such <radiative corrections>. Maintaining $m_H^2\ll\Lambda^2$ by cancelling unrelated bare and loop contributions, and readjusting the cancellation at successive orders, is the <naturalness> concern. It is different from explaining why the small scale was chosen in the first place. <Supersymmetry> primarily supplies protection against the technical instability; a complete explanation of the origin of supersymmetry-breaking scales needs additional dynamics.
The relevant comparison of mass types is:
* For a <gauge boson>, an explicit Proca mass term is forbidden by an unbroken <gauge symmetry>. This is <gauge protection of a vector mass>. In a Higgs phase, a vector mass is of order $gv$, but the stability of $v$ then depends on the <scalar mass> that determines the symmetry-breaking scale. Gauge invariance does not, by itself, solve that scalar problem.
* A chiral <fermion> mass is protected because setting it to zero restores an appropriate chiral symmetry. Perturbative corrections cannot generate a symmetry-forbidden mass; schematically $\delta m_f\propto m_f\log\Lambda$. In the <Standard Model>, chiral gauge quantum numbers prohibit a bare fermion mass, and the <Higgs mechanism> permits masses through <Yukawa couplings>. This is <chiral protection of a fermion mass>, rather than an additive correction of order $\Lambda$.
* A <squark> or <slepton> is a scalar. Its bilinear $\widetilde f^\dagger\widetilde f$ is allowed by gauge symmetries and by the ordinary chiral phase symmetry of its fermion partner. Consequently, without <supersymmetry>, those symmetries do not protect a small <scalar mass> against corrections of order $\Lambda^2$.
In exact <supersymmetry>, each <supermultiplet> has matched bosonic and fermionic degrees of freedom, with their interaction strengths related. Opposite loop signs then give <supersymmetric cancellation of quadratic divergences>. This is not just equality of state counts: the <supersymmetric relation between quartic and Yukawa couplings> is also essential. A schematic paired-loop contribution has the high-momentum form
$$
\delta m_H^2\propto |y|^2\int^{\Lambda}\frac{d^4p_E}{(2\pi)^4}
\left[\frac1{p_E^2+m_B^2}-\frac1{p_E^2+m_F^2}\right].
$$
For equal partner masses the displayed contributions cancel. With a small splitting, the difference falls as $(m_F^2-m_B^2)/p_E^4$, so the remaining ultraviolet sensitivity is logarithmic, proportional to the splitting rather than to $\Lambda^2$.
Realistic partners need not have exactly equal masses. <Soft supersymmetry breaking> permits scalar squared masses, gaugino masses and suitable trilinear interactions without restoring the unwanted quadratic sensitivity. Typically,
$$
\delta m_H^2\sim\frac{|y|^2}{16\pi^2}m_{\mathrm{soft}}^2
\log\frac{\Lambda}{m_{\mathrm{soft}}},
$$
up to coefficients and threshold details. Squarks and sleptons can therefore be heavier than their chiral fermion partners while the scalar sector remains stable against the much larger ultraviolet scale. Nevertheless, very large soft masses, particularly in the Higgs-coupled sector, leave large finite or logarithmic corrections and require tuning. This <soft scalar-mass sensitivity> remains after the quadratic divergence has cancelled. \b[Supersymmetry solves the quadratic radiative instability when breaking is suitably soft; it does not make arbitrarily heavy superpartners natural or explain the entire hierarchy by itself.]
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