The Fully-Differential Quark Beam Function at NNLO

We present the first calculation of a fully-unintegrated parton distribution (beam function) at next-to-next-to-leading order (NNLO). We obtain the fully-differential beam function for quark-initiated processes by matching it onto standard parton distribution functions (PDFs) at two loops. The fully-differential beam function is a universal ingredient in resummed predictions of observables probing both the virtuality as well as the transverse momentum of the incoming quark in addition to its usual longitudinal momentum fraction. For such double-differential observables our result provides the part of the NNLO singular cross section related to collinear initial-state radiation (ISR), and is important for the resummation of large logarithms through N$^{3}$LL