Renormalization-group and mode-coupling theories of critical dynamics

We discuss renormalization-group and mode-coupling theories of critical dynamics for several models. We show that to lowest order in $\epsilon =d_c-d$, where for $d>\mathrm{}{d}_c$ conventional theory is valid, the renormalizationgroup differential equations are related by a simple transformation to those obtained from mode-coupling theory. Furthermore, the values of the dynamical fixed-point parameters essentially coincide with the critical amplitudes of the diverging transport coefficients, as determined by mode-coupling theory. In addition, we present a correct renormalization-group treatment for dynamical systems with more than one time scale, as for the binary liquid. We stress the fact that there are two distinct classes of critical dynamics, as illustrated by the results for the models studied here. We also discuss the instability of the conventional fixed point (time-dependent Ginzburg-Landau like) against mode-coupling perturbations for $d<\mathrm{}{d}_c$, with particular emphasis on a model for superfluid helium.