Fragmentation to a jet in the large $z$ limit

We consider the fragmentation of a parton into a jet with small radius $R$ in the large $z$ limit, where $z$ is the ratio of the jet energy to the mother parton energy. In this region of phase space, large logarithms of both $R$ and $1-z$ can appear, requiring resummation in order to have a well defined perturbative expansion. Using soft-collinear effective theory, we study the fragmentation function to a jet (FFJ) in this endpoint region. We derive a factorization theorem for this object, separating collinear and collinear-soft modes. This allows for the resummation using renormalization group evolution of the logarithms $\ln R$ and $\ln(1-z)$ simultaneously. We show results valid to next-to-leading logarithmic order for the global Sudakov logarithms. We also discuss the possibility of non-global logarithms that should appear at two-loops and give an estimate of their size.