Theory of β relaxation beyond mode-coupling theory: A microscopic treatment

We develop a systematic extension of mode-coupling theory that incorporates critical dynamical fluctuations. Starting from a microscopic diagrammatic theory, we identify dominant classes of divergent diagrams near the mode-coupling transition and show that the corresponding asymptotic series dominates the mean-field prediction below an upper critical dimension d_{c}=8. To resum these divergences, we construct a mapping to a stochastic dynamical process in which the order parameter evolves under random spatiotemporal fields. This reformulation provides a controlled, fully dynamical derivation of an effective theory for β relaxation that remarkably coincides with stochastic beta relaxation theory [T. Rizzo, Long-wavelength fluctuations lead to a model of the glass crossover, Europhys. Lett. 106, 56003 (2014)10.1209/0295-5075/106/56003]. All coupling constants of the latter theory are expressed microscopically in terms of the liquid static structure factor and are computed for the paradigmatic hard-sphere system in the Percus-Yevick approximation. The analysis demonstrates that critical fluctuations alone restore ergodicity and replace the putative mean-field transition with a smooth crossover. Our results establish a predictive framework for structural relaxation beyond mean-field.