  An extension to the Soft-Collinear-Effective Theory (SCET) description of
hard jets is motivated to include the leading contributions between the
propagating partons within the jet with partons radiated from a dense extended
medium. The resulting effective Lagrangian, containing both a leading and a
power suppressed (in the hard scale $Q^2$) contribution, arises primarily from
interactions between the hard collinear modes in the jet with Glauber modes
from the medium. In this first attempt, the interactions between the hard jet
and soft and collinear partonic modes have been ignored, in an effort to focus
solely on the interactions with the Glauber modes. While the effect of such
modes on vacuum cross sections are suppressed by powers of the hard scale
compared to the terms from the SCET Lagrangian, such sub-leading contributions
are enhanced by the extent of the medium and result in measurable corrections.
The veracity of the derived Lagrangian is checked by direct comparison with
known results from full QCD calculations of two physical observables: the
transverse momentum broadening of hard jets in dense media and a reanalysis of
the transverse momentum dependent parton distribution function (TMDPDF).
