  Generalized mode-coupling theory (GMCT) constitutes a systematically
correctable, first-principles theory to study the dynamics of supercooled
liquids and the glass transition. It is a hierarchical framework that, through
the incorporation of increasingly many particle density correlations, can
remedy some of the inherent limitations of the ideal mode-coupling theory
(MCT). However, despite MCT's limitations, the ideal theory also enjoys several
remarkable successes, notably including the analytical scaling laws for the
$\alpha$- and $\beta$-relaxation dynamics. Here we mathematically derive
similar scaling laws for arbitrary-order multi-point density correlation
functions obtained from GMCT under arbitrary mean-field closure levels. More
specifically, we analytically derive the asymptotic and preasymptotic solutions
for the long-time limits of multi-point density correlators, the critical
dynamics 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-divergent
relaxation times for the two-step decay and the non-trivial relation between
their exponents are also obtained. The validity ranges of the leading-order
scaling laws are also provided by considering the leading preasymptotic
corrections. Furthermore, we test these solutions for the Percus-Yevick
hard-sphere system. We demonstrate that GMCT preserves all the celebrated
scaling laws of MCT while quantitatively improving the exponents, rendering the
theory a promising candidate for an ultimately quantitative first-principles
theory of glassy dynamics.
