  We study the quantum corrections to the polarizability of isolated metallic
mesoscopic systems using the loop-expansion in diffusive propagators. We show
that the difference between connected (grand-canonical ensemble) and isolated
(canonical ensemble) systems appears only in subleading terms of the expansion,
and can be neglected if the frequency of the external field, $\omega$, is of
the order of (or even slightly smaller than) the mean level spacing, $\Delta$.
If $\omega \ll \Delta$, the two-loop correction becomes important. We calculate
it by systematically evaluating the ballistic parts (the Hikami boxes) of the
corresponding diagrams and exploiting electroneutrality. Our theory allows one
to take into account a finite dephasing rate, $\gamma$, generated by electron
interactions, and it is complementary to the non-perturbative results obtained
from a combination of Random Matrix Theory (RMT) and the $\sigma$-model, valid
at $\gamma \to 0$. Remarkably, we find that the two-loop result for isolated
systems with moderately weak dephasing, $\gamma \sim \Delta$, is similar to the
result of the RMT+$\sigma$-model even in the limit $\omega \to 0$. For smaller
$\gamma$, we discuss the possibility to interpolate between the perturbative
and the non-perturbative results. We compare our results for the temperature
dependence of the polarizability of isolated rings to the experimental data of
Deblock \emph{et al} [\prl \ {\bf 84}, 5379 (2000); \prb \ {\bf 65}, 075301
(2002)], and we argue that the elusive 0D regime of dephasing might have
manifested itself in the observed magneto-oscillations. Besides, we thoroughly
discuss possible future measurements of the polarizability, which could aim to
reveal the existence of 0D dephasing and the role of the Pauli blocking at
small temperatures.
