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

Systems described by n-component ϕ⁴ models in a ∞^{d − 1}×L slab geometry of finite thickness L are considered at and above their bulk critical temperatureT_c,∞. The renormalization-group improved perturbation theory commonly employed to investigate the fluctuation-induced forces ("thermodynamic Casimir effect") in d = 4 − 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- expansion results which remains well behaved down to T_c,∞. The leading contributions to the critical Casimir amplitudes Δ_per and Δ_sp,spbeyond two-loop order are ∼ (u*)^{(3 − )/2}, whereu* = O() is the value of the renormalized ϕ⁴ coupling at the infrared-stable fixed point. Besides integer powers of , the small- expansions of these amplitudes involve fractional powers ^k/2, with k ⩾ 3, and powers of ln . Explicit results to order ^3/2 are presented for Δ_per and Δ_sp,sp, which are used to estimate their values at d = 3.