Why is the vacuum energy density so small yet nonzero?

In plain words

Quantum theory says empty space should be full of energy from fluctuating fields, which would curl up or blow apart the universe. The observed value is smaller than naive estimates by 44 to about 120 powers of ten, yet not zero.

Precise statement

The observed dark energy density is $\rho_{\Lambda} \sim 6 \times 10^{-30}\ \mathrm{g}/\mathrm{cm}^{3}$, i.e. $\rho_{\Lambda}^{1/4} \sim 2.3 \times 10^{-3}\ \mathrm{eV}$ in natural units, while Standard Model contributions (electroweak symmetry breaking ~ $(100\ \mathrm{GeV})^{4}$, QCD ~ $(0.2\ \mathrm{GeV})^{4}$) exceed it by $\sim 10^{55}$ and $\sim 10^{44}$, and a Planck-scale cutoff by $\sim 10^{123}$ (often quoted as $10^{120}$, using the reduced Planck mass). Find a mechanism that yields the observed value without tuning to that precision, or show that environmental selection in a multiverse is the only consistent explanation.

What would settle it

A dynamical or symmetry mechanism that predicts $\rho_{\Lambda}$ to within its observed order of magnitude, with independently testable consequences.

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