  In subleading powers of soft-collinear effective theory (SCET), the
Lagrangian contains couplings between soft quarks and hard-collinear quarks.
Matrix elements of the hard-collinear parts of these couplings are radiative
jet functions. In the position-space formulation of SCET, the Lagrangians are
constructed from gauge-invariant building blocks. Nevertheless, we find
violations of gauge invariance in the radiative jet functions that we construct
from these Lagrangians. We trace the violations of gauge invariance to Feynman
diagrams that would be present in full QCD, but are missing in the radiative
jet functions, owing to the requirement that the radiative jet functions
contain only hard-collinear lines. We construct gauge-invariant Lagrangians for
the interactions between soft quarks and hard-collinear quarks by making use of
the soft-quark equation of motion. These gauge-invariant Lagrangians are valid
for nonzero, as well as zero, quark masses. We also remark briefly on the gauge
invariance of certain Lagrangians that have been constructed in the
label-momentum formulation of SCET.

High Energy Physics - Phenomenology