Fragmentation of a jet with small radius

In this paper, we consider the fragmentation of a parton into a jet with small jet radius R. Perturbatively, logarithms of R can appear, which for narrow jets can lead to large corrections. Using soft-collinear effective theory, we introduce the fragmentation function to a jet (FFJ), which describes the fragmentation of a parton into a jet. We discuss how these objects are related to the standard jet functions. Calculating the FFJ to next-to-leading order, we show that these objects satisfy the standard Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations, with a natural scale that depends upon R. By using the standard renormalization group evolution, we can therefore resum logarithms of R. We further use the soft-collinear effective theory to prove a factorization theorem where the FFJs naturally appear, for the fragmentation of a hadron within a jet with small R. Finally, we also show how this formalism can be used to resum the ratio of jet radii for a subjet to be emitted from within a fat jet.