  Systems described by $n$-component $\phi^4$ models in a $\infty^{d-1}\times
L$ slab geometry of finite thickness $L$ are considered at and above their bulk
critical temperature $T_{c,\infty}$. The renormalization-group improved
perturbation theory commonly employed to investigate the fluctuation-induced
forces (``thermodynamic Casimir effect'') in $d=4-\epsilon$ bulk dimensions is
re-examined. It is found to be ill-defined beyond two-loop order because of
infrared singularities when the boundary conditions are such that the free
propagator in slab geometry involves a zero-energy mode at bulk criticality.
This applies to periodic boundary conditions and the special-special ones
corresponding to the critical enhancement of the surface interactions on both
confining plates. The field theory is reorganized such that a small-$\epsilon$
expansion results which remains well behaved down to $T_{c,\infty}$. The
leading contributions to the critical Casimir amplitudes
$\Delta_{\mathrm{per}}$ and $\Delta_{\mathrm{sp},\mathrm{sp}}$ beyond two-loop
order are $\sim (u^*)^{(3-\epsilon)/2}$, where $u^*=O(\epsilon)$ is the value
of the renormalized $\phi^4$ coupling at the infrared-stable fixed point.
Besides integer powers of $\epsilon$, the small-$\epsilon$ expansions of these
amplitudes involve fractional powers $\epsilon^{k/2}$, with $k\geq 3$, and
powers of $\ln \epsilon$. Explicit results to order $\epsilon^{3/2}$ are
presented for $\Delta_{\mathrm{per}}$ and $\Delta_{\mathrm{sp},\mathrm{sp}}$,
which are used to estimate their values at $d=3$.
