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The critical dynamics of relaxational stochastic models with nonconserved$n$-component order parameter $\bm{\phi}$ and no coupling to other slowvariables ("model A") is investigated in film geometries for the cases ofperiodic and free boundary conditions. The Hamiltonian $\mathcal{H}$ governingthe stationary equilibrium distribution is taken to be O(n) symmetric and toinvolve, in the case of free boundary conditions, the boundary terms$\int_{\mathfrak{B}_j}\mathring{c}_j \phi^2/2$ associated with the twoconfining surface planes $\mathfrak{B}_j$, $j=1,2$, at $z=0$ and $z=L$, wherethe enhancement variables $\mathring{c}_j$ are presumed to be subcritical orcritical. A field-theoretic RG study of the dynamic critical behavior at$d=4-\epsilon$ bulk dimensions is presented, with special attention paid to thecases where the classical theories involve zero modes at $T_{c,\infty}$. Thisapplies when either both $\mathring{c}_j$ take the critical value$\mathring{c}_{\text{sp}}$ associated with the special surface transition, orelse periodic boundary conditions are imposed. Owing to the zero modes, the$\epsilon$ expansion becomes ill-defined at $T_{c,\infty}$. Analogously to thestatic case, the field theory can be reorganized to obtain a well-definedsmall-$\epsilon$ expansion involving half-integer powers of $\epsilon$,modulated by powers of $\ln\epsilon$. Explicit results for the scalingfunctions of $T$-dependent finite-size susceptibilities at temperatures $T\geT_{c,\infty}$ and of layer and surface susceptibilities at the bulk criticalpoint are given to orders $\epsilon$ and $\epsilon^{3/2}$, respectively. Forthe case of periodic boundary conditions, the consistency of the expansions to$O(\epsilon^{3/2})$ with exact large-$n$ results is shown.

Dynamic critical behavior of model A in films: Zero-mode boundary conditions and expansion near four dimensions