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Mode-coupling theory (MCT) constitutes one of the few first-principles-basedapproaches to describe the physics of the glass transition, but the theory'sinherent approximations compromise its accuracy in the activated glassy regime.Here we show that microscopic generalized mode-coupling theory (GMCT), arecently proposed hierarchical framework to systematically improve uponstandard MCT, provides a promising pathway toward a more accuratefirst-principles description of glassy dynamics. We present a comprehensivenumerical analysis for Percus-Yevick hard spheres by performing explicitlywavenumber- and time-dependent GMCT calculations up to sixth order.Specifically, we calculate the location of the critical point, the associatednon-ergodicity parameters, the time-dependent dynamics of the densitycorrelators at both absolute and reduced packing fractions, and we test severaluniversal scaling relations in the $\alpha$- and $\beta$-relaxation regimes. Itis found that higher-order GMCT can successfully remedy some of standard MCT'spathologies, including an underestimation of the critical glass transitiondensity and an overestimation of the hard-sphere fragility. Furthermore, wenumerically demonstrate that the celebrated scaling laws of standard MCT arepreserved in GMCT at all closure levels, and that the predicted criticalexponents manifestly improve as more levels are incorporated in the GMCThierarchy. Although formally the GMCT equations should be solved up to infiniteorder to reach full convergence, our finite-order GMCT calculationsunambiguously reveal a uniform convergence pattern for the dynamics. We thusargue that GMCT can provide a feasible and controlled means to bypass MCT'smain uncontrolled approximation, offering hope for the future development of aquantitative first-principles theory of the glass transition.

Generalized mode-coupling theory of the glass transition. I. Numerical results for Percus-Yevick hard spheres