Final COMPASS results on the spin-dependent structure functions $g_1^p$ and $g_1^d$ in the deep-inelastic and nonperturbative regions

This paper summarizes the COMPASS Collaboration legacy on measurements of the proton and deuteron spin-dependent structure functions, $g_1^p$ and $g_1^d$ at $Q^2 < 1$ (GeV/c)$^2$ and $Q^2 > 1$ (GeV/c)$^2$. In both regions and at the lowest measured $x, g^d_1 (x)$ is consistent with zero while $g^p_1 (x)$ is positive. This is the first time that the spin effects are observed at such low values of $x$. The NLO QCD fit of $g_1$ world data gives well constrained quark helicity distributions; gluons are poorly determined. Quark helicity contribution to nucleon spin is $0.26 < \Delta \Sigma < 0.36$. From the COMPASS data alone the Bjorken sum rule is verified to $9\%$ accuracy and the extracted flavour-singlet axial charge is $a_0 (Q^2 = 3 (\text{GeV/}c)^2) = 0.32 \pm 0.02_{stat.} \pm 0.04_{syst.} \pm 0.05_{evol.}$.