= Solution
The <hierarchy problem> concerns the enormous separation between the <electroweak scale> and a possible fundamental ultraviolet scale such as the <reduced Planck mass>: $v\simeq246\,\mathrm{GeV}$ whereas $M_{\mathrm{Pl}}\simeq2.4\times10^{18}\,\mathrm{GeV}$. For an elementary <Higgs field>, heavy states or a physical high-energy <ultraviolet cutoff> generically generate
$$
\delta m_H^2\sim\frac{\kappa}{16\pi^2}\Lambda_{\mathrm{UV}}^2.
$$
Here $\kappa$ represents the appropriate dimensionless couplings. Maintaining a small physical Higgs mass then requires a very accurate cancellation against its bare mass parameter. The problem is sensitivity to actual high-energy physics, rather than merely the presence of a quadratic divergence in one choice of regulator.
The <cosmological constant problem> concerns the much smaller gravitationally observable <vacuum energy>. In the <Einstein equation>, a constant term in the <effective action> contributes to the <cosmological constant>, with $\Lambda_{\mathrm{eff}}=\Lambda_{\mathrm{bare}}+\rho_{\mathrm{vac}}/M_{\mathrm{Pl}}^2$ in the usual normalization. Generic zero-point contributions scale as $\Lambda_{\mathrm{UV}}^4$, and vacuum transitions also change $\rho_{\mathrm{vac}}$. The dark-energy density is only of order $10^{-47}\,\mathrm{GeV}^4\sim(\text{a few meV})^4$. The puzzle is why the total remains so small despite much larger individual contributions. A constant shift of the nongravitational <Hamiltonian> cannot remove its gravitational effect.
<Supersymmetry> relates <bosonic> and <fermionic> <degrees of freedom>, including their interactions. Their opposite loop signs cancel the quadratic contributions to a <scalar field> mass in the unbroken theory. With <soft supersymmetry breaking>, the surviving correction is controlled by the mass splitting, schematically
$$
\delta m_H^2\sim\frac{g^2}{16\pi^2}m_{\mathrm{soft}}^2\log\frac{\Lambda_{\mathrm{UV}}}{m_{\mathrm{soft}}}.
$$
Thus a sufficiently low soft-breaking scale protects a small <electroweak scale> against a much higher ultraviolet scale. This is the useful supersymmetric response to the <hierarchy problem>.
Exact global <supersymmetry> also cancels <vacuum energy>: the <supersymmetry algebra> expresses the <Hamiltonian> as a positive sum of anticommutators, so a vacuum annihilated by every <supercharge> has zero energy. Equivalently, with canonical <kinetic terms>,
$$
V=\sum_i|F_i|^2+\frac12\sum_a(D^a)^2
$$
vanishes in a <supersymmetric vacuum>. However, ordinary particles and their <superpartners> cannot have the identical mass spectrum of unbroken <supersymmetry>, so a useful phenomenological theory needs <supersymmetry breaking>. Its <vacuum energy> is generically of order the fourth power of the breaking scale; even a scale of order $10^2\,\mathrm{GeV}$ gives about $10^8\,\mathrm{GeV}^4$, enormously above the required value. In <supergravity>, a negative term can cancel positive breaking contributions:
$$
V_F=e^{K/M_{\mathrm{Pl}}^2}\left(K^{i\bar j}D_iW\,D_{\bar j}\overline W-\frac{3|W|^2}{M_{\mathrm{Pl}}^2}\right),\qquad D_iW=W_i+\frac{K_iW}{M_{\mathrm{Pl}}^2},
$$
but arranging a tiny residual is a new tuning. Consequently \b[<supersymmetry> can protect the hierarchy after soft breaking, while its exact vacuum-energy cancellation does not by itself survive realistic breaking.]
One higher-dimensional proposal uses <large extra dimensions>. Take $n$ flat compact dimensions of common radius $R$, with gravity propagating in the bulk and <Standard Model> fields confined to a four-dimensional <brane>. Reducing the gravitational <Einstein-Hilbert action> gives
$$
\frac{M_*^{n+2}}2\int d^4x\,d^ny\sqrt{-G}\,\mathcal R_{4+n}\ \longrightarrow\ \frac{M_*^{n+2}(2\pi R)^n}{2}\int d^4x\sqrt{-g}\,\mathcal R_4,
$$
so
$$
\boxed{M_{\mathrm{Pl}}^2=M_*^{n+2}(2\pi R)^n.}
$$
A fundamental gravitational scale $M_*$ near a TeV then produces apparently weak four-dimensional gravity through dilution into the large compact volume. For $n=2$ and $M_*=1\,\mathrm{TeV}$, $R=M_{\mathrm{Pl}}/(2\pi M_*^2)$ is about $0.075\,\mathrm{mm}$, using $1\,\mathrm{GeV}^{-1}\simeq1.97\times10^{-16}\,\mathrm m$. At distances much smaller than $R$, the gravitational potential scales as $r^{-(n+1)}$; at distances much larger than $R$ it scales as $r^{-1}$. Confining charged <Standard Model> fields to the <brane> avoids a light tower of charged <Kaluza-Klein modes> associated with the large radius. The primary explanatory challenge is <radius stabilization>: one must generate and stabilize $RM_*\gg1$, rather than simply postulate a huge volume and transfer the original hierarchy into a modulus.
A second proposal is the <Randall–Sundrum model>. Compactify on an interval, equivalently an orbifold with <branes> at $y=0$ and $y=L$, and use the warped metric
$$
ds^2=e^{-2k|y|}\eta_{\mu\nu}dx^\mu dx^\nu+dy^2.
$$
A negative bulk <cosmological constant> and suitably related <brane tensions> support this geometry. Integrating over the full orbifold interval $[-L,L]$ gives
$$
M_{\mathrm{Pl}}^2=M_5^3\int_{-L}^Le^{-2k|y|}\,dy=\frac{M_5^3}{k}(1-e^{-2kL}).
$$
For a <scalar field> $h$ on the <brane> at $L$, its induced metric gives the kinetic coefficient $e^{-2kL}$ and mass coefficient $e^{-4kL}m_0^2$. The canonical field is $h_c=e^{-kL}h$, hence
$$
\boxed{m_{\mathrm{phys}}=e^{-kL}m_0.}
$$
With $k$ and $M_5$ of the same fundamental order, an exponent $kL\simeq35$--$37$ converts a fundamental mass into a weak-scale mass, without requiring a large proper <compactification> volume.
Again the separation must be selected dynamically. The original warped solution leaves a massless <radion>, so merely choosing $L$ does not complete the explanation. For example, the <Goldberger–Wise stabilization mechanism> introduces a bulk <scalar field> with boundary values $v_0,v_L$ and a small mass $m_\chi$. Its static equation and solution are
$$
\chi''-4k\chi'-m_\chi^2\chi=0,\qquad \chi=Ae^{(4+\epsilon)ky}+Be^{-\epsilon ky},\qquad\epsilon=\sqrt{4+m_\chi^2/k^2}-2\simeq\frac{m_\chi^2}{4k^2}.
$$
For small backreaction and large $kL$, the leading radius-dependent potential is proportional to $e^{-4kL}(v_L-v_0e^{-\epsilon kL})^2$. Its leading minimum obeys $kL\simeq\epsilon^{-1}\log(v_0/v_L)$, so a modest boundary-value ratio and a small $\epsilon$ can give the required exponent. Corrections and the residual four-dimensional <cosmological constant> must still be controlled. \b[Both proposals need a stable dynamical origin for the <compactification> geometry; their volume or warp relations alone do not supply it.]
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