Semi-inclusive jet functions and jet substructure in JET(I) and JET(II) algorithms

Within the framework of soft collinear effective theory, we present calculations of semi-inclusive jet functions and fragmenting jet functions at next-to-leading order for both quark- and gluon-initiated jets, for jet algorithms J(ET)((I)()) and J(ET)((II)()) where one maximizes a suitable jet function. We demonstrate the consistency of the obtained results with the standard perturbative QCD calculations for the J(ET)((I)()) algorithm, while the results for fragmenting jet functions with the J(ET)((I)(I)) algorithm are new. The renonnalization-group (RG) equations for both semi-inclusive jet functions and fragmenting jet functions are derived and shown to follow the timelike Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations, independent of specific jet algorithms. The RG equation can be used to resum single logarithms of the jet size parameter beta for highly collimated jets in these algorithms where beta >> 1.