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$\beta$ and $\alpha$ relaxation processes are dynamical scaling regimes ofglassy systems occurring on two separate time scales which both diverge as theglass state is approached. We study here the crossover scaling from $\beta$- to$\alpha$- relaxation in the cooperative facilitation scenario (CFS) and showthat it is quantitatively described, with no adjustable parameter, by theleading order asymptotic formulas for scaling predicted by the mode-couplingtheory (MCT). These results establish: (i) the mutual universality of the MCTand CFS, and (ii) the existence of a purely dynamic realization of MCT which isdistinct from the well established random-first order transition scenario fordisordered systems. Some implications of the emerging kinetic-static dualityare discussed.

Crossover from $\beta$- to $\alpha$-relaxation in cooperative facilitation dynamics