  Using the spectral representation of the quark propagator we study the Dirac
decomposition of the gauge invariant quark propagator, whose imaginary part
describes the hadronization of a quark as this interacts with the vacuum.
  We then demonstrate the formal gauge invariance of the so-called jet mass,
that is of the coefficient of the chiral-odd part of the gauge invariant
propagator, that can be expressed in any gauge as the first moment of the
chiral-odd quark spectral function. This is therefore revealed to be a
\textit{bona fide} QCD observable encoding aspects of the dynamical mass
generation in the QCD vacuum, and is furthermore experimentally measurable in
specific twist-3 longitudinal-transverse asymmetries in DIS and in
semi-inclusive electron-positron collisions. In light-like axial gauges, we
also obtain a new sum rule for the spectral function associated with the gauge
fixing vector.
  We finally present a gauge-dependent formula that connects the second moment
of the chiral-even coefficient of the quark spectral function to invariant mass
generation and final state rescattering in the hadronization of a quark.
Finding twist-4 experimental observables sensitive to this quantity is left for
future work.
