Time-Like Evolution

Due to confinement in QCD we can not observe partons, such as quarks and gluons, directly in particle collider experiments. Instead, stable hadrons are detected which originate from parton interactions.

The fragmentation functions (FF) encode the information on the probability for a hadron carrying a specified momentum fraction to ‘fragment’ from a given parton. These functions are non-perturbative and usually require a global QCD analysis of experimental data involving different processes for their reliable determination. This makes the FF similar to PDF as both rely on similar factorization theorems and, thus, on similar RGE. In practice, the relevant Feynman diagrams can indeed be related by a crossing symmetry which in turn means certain Mandelstam variables become for FF time-like instead of space-like. The relevant setting in the operator card is thus called time_like = True.

We implement the time-like DGLAP anomalous dimensions up to NNLO in time_like. The implementation for the LO and NLO splitting functions is based on [GRV93, MM06] and the implementation for the NNLO splitting functions is based on [AMV12, MMV06, MV08]. We also implement the time-like matching conditions up to NLO in time_like which are based on [CNO05]. The time-like matching conditions for NNLO are not known. Supplying new anomalous dimensions and new matching conditions is the only change required for the eko program (e.g. the solution strategies are unaffected).

The time-like evolution has been benchmarked using selected FF sets from lhapdf. In addition, the splitting functions have been tested against functions from MELA, which is an alternative Mellin Space evolution code.