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 pT spectrum, etc. We show that they may directly be calculated as phase-space integrals of QCD splitting functions. At 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 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 which are computed using differential equations.