Quark beam function at next-to-next-to-next-to-leading order in perturbative QCD in the generalized large-$N_c$ approximation

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 ${\mathrm{N}}^3\mathrm{LO}$ in perturbative QCD is within reach.

Figure 1

Top: example of a triple-real emission amplitude with a quark-antiquark pair in the final state which contributes to the bare beam function in the leading-color approximation and therefore has been included in our computation. Bottom: example of a similar amplitude which is subleading in $N_c$ and therefore is not included in our computation. The box labeled $\mathcal{H}$ denotes the hard scattering process.