  We present the matching coefficient for the quark beam function at
next-to-next-to-next-to-leading order in perturbative QCD in the generalized
large $N_c$-approximation, $N_c \sim N_f \gg 1$. Although several refinements
are still needed to make this result interesting for phenomenological
applications, our computation shows that a fully-differential description of
simple color singlet production processes at a hadron collider at N$^3$LO in
perturbative QCD is within reach.
