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Systems described by $n$-component $\phi^4$ models in a $\infty^{d-1}\timesL$ slab geometry of finite thickness $L$ are considered at and above their bulkcritical temperature $T_{c,\infty}$. The renormalization-group improvedperturbation theory commonly employed to investigate the fluctuation-inducedforces (``thermodynamic Casimir effect'') in $d=4-\epsilon$ bulk dimensions isre-examined. It is found to be ill-defined beyond two-loop order because ofinfrared singularities when the boundary conditions are such that the freepropagator in slab geometry involves a zero-energy mode at bulk criticality.This applies to periodic boundary conditions and the special-special onescorresponding to the critical enhancement of the surface interactions on bothconfining plates. The field theory is reorganized such that a small-$\epsilon$expansion results which remains well behaved down to $T_{c,\infty}$. Theleading contributions to the critical Casimir amplitudes$\Delta_{\mathrm{per}}$ and $\Delta_{\mathrm{sp},\mathrm{sp}}$ beyond two-looporder are $\sim (u^*)^{(3-\epsilon)/2}$, where $u^*=O(\epsilon)$ is the valueof the renormalized $\phi^4$ coupling at the infrared-stable fixed point.Besides integer powers of $\epsilon$, the small-$\epsilon$ expansions of theseamplitudes involve fractional powers $\epsilon^{k/2}$, with $k\geq 3$, andpowers of $\ln \epsilon$. Explicit results to order $\epsilon^{3/2}$ arepresented for $\Delta_{\mathrm{per}}$ and $\Delta_{\mathrm{sp},\mathrm{sp}}$,which are used to estimate their values at $d=3$.Comment: 7 pages, latex with epl stylefile

Fluctuation-induced forces in periodic slabs: Breakdown of epsilon expansion at the bulk critical point and revised field theory