Fragmentation in jets at NNLO

Beam and jet functions in soft-collinear effective theory describe collinear initial- and final-state radiation (jets), and enter in factorization theorems for $N$-jet production, the Higgs $p_T$ spectrum, etc. We show that they may directly be calculated as phase-space integrals of QCD splitting functions. At next-to-leading order (NLO) all computations are trivial, as we demonstrate explicitly for the beam function, the transverse-momentum-dependent beam function, the jet function, and the fragmenting jet function. This approach also highlights the role of crossing symmetry in these calculations. At next-to-next-to leading order (NNLO) we reproduce the quark jet function and calculate the fragmenting quark jet function for the first time. Here we use two methods: a direct phase-space integration and a reduction to master integrals that are computed using differential equations.