We connect at the operator level the inclusive gauge-invariant quark propagator and the quark to single hadron fragmentation correlator in QCD. Exploiting a new spectral decomposition for the quark propagator, we then derive a complete set of momentum sum rules for the quark fragmentation functions into a single unpolarized hadron up to twist-three. Known results are recovered, and new sum rules discovered.
The spectral decomposition of the jet correlator also allows one to define a gauge-invariant dressed quark (or "jet") mass. Remarkably, the mass sum rules for the $E$ and $\tilde E$ twist-3 fragmentation functions provide an experimental gateway to this jet mass and its interaction-dependent component, which can thus be potentially measured in deep-inelastic processes and shed light on the dynamical generation of mass in QCD.