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Generalized mode-coupling theory (GMCT) constitutes a systematicallycorrectable, first-principles theory to study the dynamics of supercooledliquids and the glass transition. It is a hierarchical framework that, throughthe incorporation of increasingly many particle density correlations, canremedy some of the inherent limitations of the ideal mode-coupling theory(MCT). However, despite MCT's limitations, the ideal theory also enjoys severalremarkable successes, notably including the analytical scaling laws for the$\alpha$- and $\beta$-relaxation dynamics. Here we mathematically derivesimilar scaling laws for arbitrary-order multi-point density correlationfunctions obtained from GMCT under arbitrary mean-field closure levels. Morespecifically, we analytically derive the asymptotic and preasymptotic solutionsfor the long-time limits of multi-point density correlators, the criticaldynamics with two power-law decays, the factorization scaling laws in the$\beta$-relaxation regime, and the time-density superposition principle in the$\alpha$-relaxation regime. The two characteristic power-law-divergentrelaxation times for the two-step decay and the non-trivial relation betweentheir exponents are also obtained. The validity ranges of the leading-orderscaling laws are also provided by considering the leading preasymptoticcorrections. Furthermore, we test these solutions for the Percus-Yevickhard-sphere system. We demonstrate that GMCT preserves all the celebratedscaling laws of MCT while quantitatively improving the exponents, rendering thetheory a promising candidate for an ultimately quantitative first-principlestheory of glassy dynamics.

Generalized mode-coupling theory of the glass transition. II. Analytical scaling laws