  The jet fragmentation function describes the longitudinal momentum
distribution of hadrons inside a reconstructed jet. We study the jet
fragmentation function in proton-proton collisions in the framework of
soft-collinear effective theory (SCET). We find that, up to power corrections,
the jet fragmentation function can be expressed as the ratio of the fragmenting
jet function and the unmeasured jet function. Using renormalization group
techniques, we are able to resum large logarithms of jet radii R in the
perturbative expansion of the cross section. We use our theoretical formalism
to describe the jet fragmentation functions for light hadron and heavy meson
production measured at the Large Hadron Collider (LHC). Our calculations agree
very well with the experimental data for the light hadron production. On the
other hand, although our calculations for the heavy meson production inside
jets are consistent with the PYTHIA simulation, they fail to describe the LHC
data. We find that the jet fragmentation function for heavy meson production is
very sensitive to the gluon-to-heavy-meson fragmentation function.
