Parton Fragmentation within an Identified Jet at NNLL

The fragmentation of a light parton i to a jet containing a light energetic hadron h, where the momentum fraction of this hadron as well as the invariant mass of the jet is measured, is described by "fragmenting jet functions". We calculate the one-loop matching coefficients J_{ij} that relate the fragmenting jet functions G_i^h to the standard, unpolarized fragmentation functions D_j^h for quark and gluon jets. We perform this calculation using various IR regulators and show explicitly how the IR divergences cancel in the matching. We derive the relationship between the coefficients J_{ij} and the quark and gluon jet functions. This provides a cross-check of our results. As an application we study the process e+ e- to X pi+ on the Upsilon(4S) resonance where we measure the momentum fraction of the pi+ and restrict to the dijet limit by imposing a cut on thrust T. In our analysis we sum the logarithms of tau=1-T in the cross section to next-to-next-to-leading-logarithmic accuracy (NNLL). We find that including contributions up to NNLL (or NLO) can have a large impact on extracting fragmentation functions from e+ e- to dijet + h.