Two-loop anomalous dimensions for small-R jet versus hadronic fragmentation functions

We study the collinear fragmentation of highly energetic jets defined with a small jet radius. In particular, we investigate how the corresponding fragmentation functions differ from their hadronic counterpart defined in the common MS scheme. Contrary to previous assumptions in the literature, we find that the anomalous dimensions governing the perturbative evolution of the two fragmentation functions differ starting at the two loop order. We derive for the first time the new anomalous dimensions at two loops and confirm our predictions by comparing the inclusive small-R jet spectrum against a fixed order perturbative calculation at O(α_s^2). To investigate the dependence of the anomalous dimension on the kinematic cutoff variable, we also study the fragmentation functions of Cambridge jets defined with a transverse momentum cutoff as to an angular cutoff R. In this case we find that the two-loop anomalous dimensions coincide with the MS DGLAP ones, highlighting a correspondence between the MS scheme and a transverse-momentum cutoff.